Let p be an odd prime, and let k be a nonzero nature number. Suppose that nonabelian group G is a central extension as follows \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \to G\prime \to G \to {{\mathbb{Z}}_{{p<^>k}}} \times \cdots \times {{\mathbb{Z}}_{{p<^>k}}},$$\end{document} where G ' approximately equal to DOUBLE-STRUCK CAPITAL Z(p)k, and zeta G/G ' is a direct factor of G/G '.Then G is a central product of an extraspecial p(k)-group E and zeta G. Let divide E divide = p((2n+1)k) and divide zeta G divide = p((m+1)k). Suppose that the exponents of E and zeta G are p(k+l) and p(k+r), respectively, where 0 <= l, r <= k. Let Aut(G ')G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G ', let Aut(G/zeta G,zeta G)G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center zeta G, and let Aut(G/zeta G,zeta G/G ')G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on zeta G/G '. Then (i) The group extension 1 -> Aut(G ')G -> Aut G -> Aut G ' -> 1 is split. (ii) Aut(G ')G/Aut(G/zeta G,zeta G)G approximately equal to G(1) x G(2), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p<^>k}}})\ltimes H \le {G_1} \le {\rm{Sp}}(2n,{{\bf{Z}}_{{p<^>k}}})$$\end{document}, H is an extraspecial p(k) -group of order p((2n-1)k) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\rm{GL}}(m - 1,{{\bf{Z}}_{{p<^>k}}})\ltimes \mathbb{Z}_{{p<^>k}}<^>{(m - 1)})\ltimes \mathbb{Z}_{{p<^>k}}<^>{(m)} \le {G_2} \le {\rm{GL}}(m,{{\bf{Z}}_{{p<^>k}}})\ltimes \mathbb{Z}_{{p<^>k}}<^>{(m)}$$\end{document}. In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_1} = {\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p<^>k}}})\ltimes H$$\end{document} if and only if l = k and r = 0; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_1} = {\rm{Sp}}(2n,{{\bf{Z}}_{{p<^>k}}})$$\end{document} if and only if l <= r; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_2} = ({\rm{GL}}(m - 1,{{\bf{Z}}_{{p<^>k}}})\ltimes {\mathbb{Z}}_{{p<^>k}}<^>{(m - 1)})\ltimes {\mathbb{Z}}_{{p<^>k}}<^>{(m)}$$\end{document} if and only if r = k; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_2} = {\rm{GL}}(m,{{\bf{Z}}_{{p<^>k}}})\ltimes {\mathbb{Z}}_{{p<^>k}}<^>{(m)}$$\end{document} if and only if r = 0. (iii) Aut(G ')G/Aut(G/zeta G,zeta G/G ')G approximately equal to G(1) x G(3), where G(1) is defined in (ii); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{GL}}(m - 1,{{\bf{Z}}_{{p<^>k}}})\ltimes {\mathbb{Z}}_{{p<^>k}}<^>{(m - 1)} \le {G_3} \le {\rm{GL}}(m,{{\bf{Z}}_{{p<^>k}}})$$\end{document}. In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_3}{\rm{= GL}}(m - 1,{{\bf{Z}}_{{p<^>k}}})\ltimes {\mathbb{Z}}_{{p<^>k}}<^>{(m - 1)}$$\end{document} if and only if r = k; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_3} = {\rm{GL}}(m,{{\bf{Z}}_{{p<^>k}}})$$\end{document} if and only if r = 0. (iv) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G/G\prime}}G \cong {\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G\rtimes {\mathbb{Z}}_{{p<^>k}}<^>{(m)}$$\end{document}. If m = 0, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G = {\rm{Inn}}\,G \cong \mathbb{Z}_{{p<^>k}}<^>{(2n)}$$\end{document}; If m > 0, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G \cong \mathbb{Z}_{{p<^>k}}<^>{(2nm)} \times \mathbb{Z}_{{p<^>{k - r}}}<^>{(2n)}$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G/{\rm{Inn}}\,G \cong \mathbb{Z}_{{p<^>k}}<^>{(2n(m - 1))} \times \mathbb{Z}_{{p<^>{k - r}}}<^>{(2n)}$$\end{document}.
