Maps on spaces of symmetric matrices preserving idempotence

Suppose F is an arbitrary field. Let |F| be the number of the elements of F. Let M_n(F) be the space of all n×n matrices over F, and let S_n(F) be the subset of M_n(F) consisting of all symmetric matrices. Let V{S_n(F),M_n(F)}, a map Φ:V→V is said to preserve idempotence if A-λB is idempotent if and only if Φ(A)-λΦ(B) is idempotent for any A,BV and λF. It is shown that: when the characteristic of F is not 2, |F|>3 and n4, Φ:S_n(F)→S_n(F) is a map preserving idempotence if and only if there exists an invertible matrix PM_n(F) such that Φ(A)=PAP^-1 for every AS_n(F) and P^tP=aI_n for some nonzero scalar a in F.