Let $R$ be a ring of positive characteristic. Let $G$ be a commutative affine and smooth group scheme over $R$. I consider two abelian groups:$$M(G):=\mathrm{Hom}(G,\mathbb{G}_{a,R})$$and$$N(G):=\mathrm{Hom}(\mathbb{G}_{a,R},G)$$of group scheme homomorphisms over $R$, where $\mathbb{G}_{a,R}$ denote the additive groupe scheme over $R$. Both groups are module over $R$ through its map to $\mathrm{End}(\mathbb{G}_{a,R})$.
The formation of $M$ commutes with flat base change (e.g. one may deduce this from remark 3.3 in Hartl - Isogenies of abelian Anderson A-modules and A-motives ). However, this is not true for $N$ as @afh pointed out in his/her answer.
Still, does $N$ commute with flat base change in either of these scenarii?
- if $G$ is a form of $\mathbb{G}_a^d$? [meaning that there exists a faithfully flat ring homomorphism $R\to R'$ for which $G\times_R R'$ is isomorphic to $\mathbb{G}_{a,R'}^d$ as a group scheme];
- if the flat base change $R\to S$ is such that $S$ is perfect.