# Terminological question on "action factors through"

What does it mean that the action of a group on some space factors through the action of another one?

The action of a group $G$ on $X$ (in particular a set, but it could be an object in any category, for instance a topological space) can be encoded as a group homomorphism $G\to{\rm Aut}(X)$.
In group theory we say a homomorphism $G\xrightarrow{\phi} A$ "factors through $H$" if there exists more group homomorphisms $G\xrightarrow{\psi}H$ and $H\xrightarrow{\theta}A$ such that their composition is $G\xrightarrow{\phi}A$, i.e. $\phi=\theta\circ\psi$; we say the homomorphism $\phi$ factors through $\theta$ or $\psi$ if one of those is specified as well. Conceptually, we can think of "$U$ factoring through $V$" as $U$ being decomposed into $V$ plus something else, which for morphisms means we decompose into maps under composition.
Thus the action of a group $G$ factors through the action of a group $H$ if the corresponding homomorphism $G\to{\rm Aut}(X)$ factors through the homomorphism $H\to{\rm Aut}(X)$.
If the other group is a quotient group of the first $G$, it means that the action can be written as composition of the canonical projection (group morphism) from $G$ to the quotient, followed by the action of the quotient (group morphism to a general linear group). It implies that the kernel of the projection acts trivially.