This has been more or less said in the comments, but it seems like someone should write an answer.

If $H$ is a subgroup of $G$ and $H$ has a left action on a space $X$, then a space $G\times _HX$ is defined, a quotient space of $G\times X$, by declaring $(gh,x)\sim (g,hx)$. (Note that we are using the right action of $H$ on the space $G$. If we like, we can create a left action on $G\times X$ using the right action on $G$ and the left action on $X$, writing $h(g,x)= (gh^{-1},hx)$. But this is unnecessary.)

Let's denote the equivalence class of $(g,x)$ by, say, $[g,x]$. The quotient space gets a left $G$-action, defined by $g_1[g_2,x]=[g_1g_2,x]$. (This is well-defined because the left action of $G$ on (the space) $G$ commutes with the right action: $[g_1(g_2h),x]=[(g_1g_2)h,x]=[g_1g_2,hx]$.

In categorical terms: The subgroup $H\subset G$ (or the homomorphism $H\to G$) is giving us a functor $X\mapsto G\times_HX$ from the category of spaces with left $H$-action to the category of spaces with left $G$-action. This functor is left adjoint to the forgetful functor.

This is very similar to what’s happens when a homomorphism $R\to S$ of (associative) rings gives a functor $X\mapsto S\otimes_R X$ from $R$-modules to $S$-modules, again left adjoint to the forgetful functor.

This has been more or less said in the comments, but it seems like someone should write an answer.

If $H$ is a subgroup of $G$ and $H$ has a left action on a space $X$, then a space $G\times _HX$ is defined, a quotient space of $G\times X$, by declaring $(gh,x)\sim (g,hx)$. Note that we are using the right action of $H$ on the space $G$. (If we like, we can create a left action on $G\times X$ using the right action on $G$ and the left action on $X$, writing $h(g,x)= (gh^{-1},hx)$. But this is unnecessary.)

Let's denote the equivalence class of $(g,x)$ by, say, $[g,x]$. The quotient space gets a left $G$-action, defined by $g_1[g_2,x]=[g_1g_2,x]$. (This is well-defined because the left action of $G$ on (the space) $G$ commutes with the right action: $[g_1(g_2h),x]=[(g_1g_2)h,x]=[g_1g_2,hx]$.

In categorical terms: The subgroup $H\subset G$ (or the homomorphism $H\to G$) is giving us a functor $X\mapsto G\times_HX$ from the category of spaces with left $H$-action to the category of spaces with left $G$-action. This functor is left adjoint to the forgetful functor.

This is very similar to how a homomorphism $R\to S$ of (associative) rings gives a functor $X\mapsto S\otimes_R X$ from $R$-modules to $S$-modules, again left adjoint to the forgetful functor.