Linear maps preserving inverses of matrices on symmetric matrix space

Suppose F is a field of characteristic not 2 and |F|3, m and n are positive integers, Let Sn(F) and Mm(F) be the vector spaces of all n×n symmetric matrices and all m×m full matrices over F, respectively. Let GLn(F) be the set of all n×n nonsingular matrices. A linear map f from Sn(F) to Mm(F) is said to inverse-preserving if f(X)-1=f(X-1) for every X∈Sn(F)∩GLn(F). Linear maps preserving inverses of matrices from Sn(F) to Mm(F)(respectively Sm(F))are characterized.