ca.analysis and odes - Name for topology making group action continuous

Fix a set $X$ with right $G$-action. Give $X$ a topology $tau$ and make $G$ a topological group. (These topologies need not make the action continuous).

We can define another topology $tau'$ on $X$ as the largest topology making the action $(X,tau) times G to (X,tau')$ continuous. (This is also called the quotient topology on $X$ with respect to the action $(X,tau) times G to X$.)

Note that if the $G$-action is continuous for $tau$ then $tau'= tau$.

For example, if $X = mathbb{R}$, $tau$ is the discrete topology and $G$ is $(mathbb{R}, +)$ with the usual topology acting on $X$ by addition, then $G times X / sim = mathbb{R}$ with the usual topology (unless I am much mistaken).

More interesting examples exist, e.g. the Skorokhod topology (again unless I am mistaken).

This construction feels useful enough that it must be well known and have a name. Can anyone provide me with more information?

[EDIT: actually I don't think it's necessary that $G$ is a topological group, just that it's a group with a topology. Although it is probably necessary for inversion to be continuous at the identity and for multiplication to be continuous on ${e} times G$.]

[EDIT: made the presentation clearer to address the existing comments, changed title]