Let $G$ be a group acting on a Noetherian scheme $X$ over $\mathbb{C}$ and $\mathcal{E}$ be a $G$-equivariant coherent sheaf on $X$ where the support of $\mathcal{E}$ contains the fixed locus $X^G$. I have the following questions
1)Is the restriction of $\mathcal{E}$ to the fixed locus $X^G$ a locally free sheaf of constant rank?
2)If the first question is not true, what conditions should one add to get a locally free sheaf of constant rank on the restriction to $X^G$ out of a coherent $G$-equivariant sheaf?
I edited the question, original version did not have any restriction on the support and I wasn't asking the second question.