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)$ ?