Name of a category constructed from the action of a group on a category

Let $G$ be a group acting on a category $C$. That is we have a morphism of groups $G \to Aut(C)$.

We can now form a new category as follows:

• Its objects are tuples consisting of an object $x$ of $C$, together with morphisms $\Phi_g : g \cdot x \to x$, satisfying the condition : $\Phi_g \circ (g \cdot \Phi_h) = \Phi_{gh}$
• A morphism $(x,\Phi) \to (y,\Psi)$ is a morphism $F:x \to y$ such that $\Psi_g \circ (g \cdot F) = F \circ \Phi_g$

Does this new category (or at least its objects) have a name?

Yes. It's called the category of homotopy fixed points $C^G$ for the action of $G$ on $C$. (You need a unit condition on $\Phi_e$ but that's not a big deal.) See, for example, the sequence of blog posts starting here.
A typical example is that $G = \text{Gal}(L/K)$ is a Galois group and $C$ is the category of objects of some kind "over $L$" (e.g. $L$-vector spaces, $L$-algebras, $L$-schemes); then Galois descent, when it holds, asserts that $C^G$ is the corresponding category of objects "over $K$." (Most resources about Galois descent on the internet will go to great lengths to avoid saying this.)
Note that $\text{Aut}(C)$ is a 2-group, not a group, and this distinction really matters. It's possible for $\pi_0(\text{Aut}(C))$ to be trivial (meaning every equivalence $C \to C$ is equivalent to $\text{id}_C$) but for $\pi_1(\text{Aut}(C))$ to be interesting, in which case actions of $G$ on $C$ can still be interesting even though the individual functors associated to each $g \in G$ are equivalent to the identity. The correct definition of an action of a group on a category is therefore slightly more involved and this slightly modifies the definition of a homotopy fixed point.