On maps preserving square roots of idempotent and rank-one nilpotent matrices

We characterize bijective linear maps on M-n(C) that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions and a map preserving square-zero matrices. Next., we consider bijective linear maps that preserve the square roots of a rank-one nilpotent matrix. These maps do not have standard forms when compared to similar linear preserver problems.