Let M be a right module over a ring R. In this manuscript,
we shall study on a special case of F-inverse split modules
where F is a fully invariant submodule of M introduced in .
We say M is Z
2(M)-inverse split provided f^(-1)(Z2(M)) is a direct
summand of M for each endomorphism f of M. We prove that M
is Z2(M)-inverse split if and only if M is a direct sum of Z2(M)
and a Z2-torsionfree Rickart submodule. It is shown under some
assumptions that the class of right perfect rings R for which every
right R-module M is Z2(M)-inverse split (Z(M)-inverse split) is
precisely that of right GV-rings.