# Could we characterize injective objects in the category of $G$-equivariant sheaves?

Let $$k$$ be a field and $$X$$ be a topological space. We consider Sh$$(X)$$, the category of sheaves of $$k$$-vector spaces on $$X$$.

Let $$G$$ be a topological group which act on $$X$$ continuously from the left. Consider the simplicial space $$[G\backslash X]_{\cdot}$$ where $$[G\backslash X]_n=\underbrace{G\times \ldots \times G}_{n\text{ copies of }G\text{'s}}\times X$$ with structural maps $$d_0(g_1,\ldots, g_n,x)=(g_2,\ldots,g_n,g_1^{-1}x);$$ $$d_i(g_1,\ldots, g_n,x)=(g_1,\ldots, g_ig_{i+1},\ldots, g_n,x), ~1\leq i\leq n-1;$$ $$d_n(g_1,\ldots, g_n,x)=(g_1,\ldots, g_{n-1},x);$$ and $$s_0(g_1,\ldots, g_n,x)=(e,g_1,\ldots, g_n,x);$$ $$s_i(g_1,\ldots, g_n,x)=(g_1,\ldots, g_i,e , g_{i+1},\ldots, g_nx),~1\leq i\leq n-1;$$ $$s_n(g_1,\ldots, g_n,x)=(g_1,\ldots, g_n,e,x).$$

A $$G$$-equivariant sheaf on $$X$$ is a pair $$(\mathcal{F},\theta)$$, where $$\mathcal{F}\in \text{Sh}(X)$$ and $$\theta$$ is an isomorphism $$\theta: d_0^*\mathcal{F}\overset{\sim}{\to} d_1^*\mathcal{F},$$ satisfying the cocycle condition $$d_2^*\theta\circ d_0^*\theta=d_1^*\theta, \text{ and } s_0^*\theta=\text{id}_{\mathcal{F}}.$$ We denote the category of $$G$$-equivariant sheaves on $$X$$ by $$\text{Sh}_G(X)$$. It is clear that $$\text{Sh}_G(X)$$ is an abelian category and the forgetful functor For$$: \text{Sh}_G(X)\to \text{Sh}(X)$$ is exact.

I am interested in the injective objects in $$\text{Sh}_G(X)$$. We know that in general, $$\text{For}: \text{Sh}_G(X)\to \text{Sh}(X)$$ does not preserve injective objects, unless $$G$$ is discrete.

Here are my questions

In the case that $$G$$ is discrete, $$\mathcal{F}$$ is injective as a sheaf is a necessary but not sufficient condition that $$(\mathcal{F},\theta)$$ is an injective object in $$\text{Sh}_G(X)$$. Then what are additional requirements to make sure that $$(\mathcal{F},\theta)$$ is injective?

In then general case that $$G$$ is not discrete, what are injective objects in $$\text{Sh}_G(X)$$?