Let $R=\underleftarrow\lim (R/\mathfrak m^i)$ be a complete local ring, with residue field $k=R/\mathfrak m$, and let's assume that $R$ is Noetherian.

If $R$ is a $k$-algebra, then I believe that the following is correct:

The injective hull of $k$ can be described as the set of continuous homomorphisms $$ E(k) = \hom_{\mathrm{cts}}(R,k)=\underrightarrow\lim (\hom(R/\mathfrak m^i,k)), $$ where continuity is with respect to the $\mathfrak m$-adic topology.

Let me now assume that we're in mixed characteristic ($R$ is no longer a $k$-algebra).

**Question:** Is there a description of $E(k)$ along the same lines as above?

In particular, does the automorphism group of $R$ act on $E(k)$?