Linear maps preserving group inverses of upper triangular matrices over fields

Suppose F is a field with at least four elements, and n≥2 is a positive integer. Let M_ n(F) and T_ n(F) be the linear spaces of n×n full matrices and upper triangular matrices over F, respectively. All linear injective maps from T_ n(F) to M_ n(F) preserving group inverses of matrices are first characterized, and thereby all linear bijective maps from T_ n(F) to itself preserving group inverses of matrices are characterized.