# Does the category of $G$-equivariant sheaves have enough injectives?

The question is related to this one.

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 $$d_2^*\theta\circ d_0^*\theta=d_1^*\theta, \text{ and } s_0^*\theta=\text{id}_{\mathcal{F}}.$$

A morphism $$\phi: (\mathcal{F},\theta)\to (\mathcal{G},\xi)$$ is a sheaf morphism $$\phi: \mathcal{F}\to \mathcal{G}$$ that commute with $$\theta$$ and $$\xi$$. 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.

An equivariant sheaf $$(\mathcal{I},\eta)$$ is called injective if for any monomorphsim of equivariant sheaves $$\phi: (\mathcal{F},\theta)\to (\mathcal{G},\xi)$$ and any morphism of equivariant sheaves $$\psi: (\mathcal{F},\theta)\to (\mathcal{I},\eta)$$, $$\psi$$ could be extended to a morphism $$\chi: (\mathcal{G},\xi)\to (\mathcal{I},\eta)$$ such that $$\chi\circ \phi=\psi$$.

My question is: does $$\text{Sh}_G(X)$$ have enough injective objects? In other words, for any object $$(\mathcal{F},\theta)\in \text{Sh}_G(X)$$, does there exist an injective object $$(\mathcal{I},\eta)$$ together with a monomorphism $$\phi: (\mathcal{F},\theta)\to (\mathcal{I},\eta)$$?

I am not sure if this question is the same as the existence of enough injectives in the category of sheaves on stacks.