Linear k-power/k-potent preservers between matrix spaces

Suppose F is a field. Let Mn(F) be the linear space of all n × n matrices over F, and let Sn(F) be its subspace consisting of all symmetric matrices. Let m, n, k be positive integers with k ⩾ 2, and let V∈{Mn(F),Sn(F)} and W∈{Mm(F),Sm(F)}. A linear map f:V→W is called a k-power preserver if f(A)k = f(Ak) for every A∈V, and a k-potent preserver if f(A)k = f(A) for any A∈V with Ak = A. We characterize: (I) linear k-power preservers from Mn(F) to Mm(F) when ch F > k or ch F = 0; (II) linear k-potent preservers from Mn(F) to Mm(F) when F is an algebraically closed field with ch F = 0; (III) linear k-power preservers from Sn(F) to Mm(F) (respectively, Sm(F)) when ch F > k ⩾ 5 or ch F = 0; and (IV) linear k-potent preservers from Sn(F) to Mm(F) (respectively, Sm(F)) when F is an algebraically closed field with ch F = 0.