Let $G$ is affine algebraic group over $\mathbb{C}$ acting on a smooth scheme $X$ over $\mathbb{C}$, let $\mathcal{F},\mathcal{G}$ be two quasi coherent equivariant sheaves on $X$. Is there a natural equivariant structure we can put on the sheaf hom $\mathcal{Hom}(\mathcal{F},\mathcal{G})$?
This is what I think it should be, if $p:G\times X\to X$ and $a:G\times X\to X$ denotes the projection and action map respectively then in if $\mathcal{F}$ is coherent and the action map is flat we have isomorphisms
$$p^*\mathcal{Hom}(\mathcal{F},\mathcal{G})\cong \mathcal{Hom}(p^*\mathcal{F},p^*\mathcal{G})\cong \mathcal{Hom}(a^*\mathcal{F},a^*\mathcal{G})\cong a^*\mathcal{Hom}(\mathcal{F},\mathcal{G})$$ which I think we can use take as our equivariant structure on $\mathcal{Hom}(\mathcal{F},\mathcal{G})$. But, what do we do if $\mathcal{F}$ is not coherent or the action map is not flat?
I don't know if that's helpful, but a few remarks: I think the action map $a:G\times X \rightarrow X$ is always flat for any affine algebraic group see here this paper p.4. They also explain equivariance.
If you're dealing with coherent sheaves, you should consider the category $\mathcal{D}^b(X)^{G}$ of $G$-equivariant coherent sheaves. Note that the isomorphism $\theta: a^*\mathcal{F}\cong p^*\mathcal{F}$ IS the equivariant structure and should not be ignored. For simplicity: If $\mathcal{F}$ is a complex vector bundle of rank $n$ then $\theta$ is a linearisation of the fibres $\theta_x:\mathcal{F_x}\rightarrow F_{g(x)}$, i.e. an object in $GL_n(\mathbf{C})$!
Concerning your question of pullback: If you go to the derived category and you have smooth projective varieties $f:X \rightarrow Y$, every $\mathcal{F}$ admits a locally free resolution, and locally free's commute with (derived) pullback, i.e. $Lf^*\mathcal{RH}om(\mathcal{F},\mathcal{E}) \cong \mathcal{RH}om(Lf^*\mathcal{F},Lf^*\mathcal{E})$ for complexes $\mathcal{F},\mathcal{E} \in \mathbf{D}^b(Y)$.