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.