FINITE GROUPS OF EVEN ORDERS ALL OF WHOSE FOURTH MAXIMAL SUBGROUPS ARE WEAKLY s2-PERMUTABLE

Let p be a prime number. A positive integer m is said to be a p'-number provided p inverted iota m. Let G be a finite group, and let H be a subgroup of G. We say that H is weakly s(p)-permutable in G if G has a subnormal subgroup K such that G = HK, H-sG <= K and vertical bar H boolean AND K : H-sG vertical bar is a p'-number, where H-sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We study the structure of a finite group G of even order all of whose fourth maximal subgroups are weakly s(2)-permutable in G.