Inverse-preserving linear maps between spaces of matrices over fields

Suppose F is a field different from F-2, the field with two elements. Let M-n(F) and S-n(F) be the space of n x n full matrices and the space of n x n symmetric matrices over F, respectively. For any G(1), G(2) is an element of {S-n(F), M-n(F)}, we say that a linear map f from G(1) to G(2) is inverse-preserving if f(X)-1 = from G(1) to G(2). In this paper the sets L(S-n(F),M-n(F)), L(S-n(F),S-n(F)), L(M-n(F),M-n(F)) and L(M-n(F),S-n(F)) are characterized.