Linear preservers on group inverses of symmetric matrices over field of characteristic 2

Suppose F is a field of characteristic 2 and n≥2 is a positive integer.Let M n(F) and S n(F) be the linear spaces of n×n full matrices and symmetric matrices over F,respectively.We first characterize all linear injective maps from M n(F) to S n(F) preserving group inverses of matrices,and thereby all linear bijective maps from S n(F) to itself preserving group inverses of matrices are characterized.