Can prime fields act two ways on the same abelian group?

[imurme8]

A problem asks to find an abelian group V and a field F such that there exist two different actions, call them \cdot and \odot, of F on V such that V is an F-module.

A usual way to solve this is to take any vector space over a field with a non-trivial automorphism group, and define r\odot \mu to be f(r)\cdot \mu for f\in \text{Aut}(F), f\neq \iota.

My question is: is this essentially the only way? Given two different actions of F on V, can we construct a non-trivial automorphism of F?

Or, a weaker result: can we show that a prime field cannot act in two different ways on a module?

 

[fresh_42]

We always have the trivial operation: $F.V \equiv 0$.
Another example are direct versus semidirect products. And if the automorphism group is large enough, we can have different conjugations.

And $D_{12} \cong V_4 \ltimes \mathbb{Z}_3 \cong D_6 \times \mathbb{Z}_2$ might provide an example as we have two copies of $\mathbb{Z}_2$ here with a different operation on the rest.