I've posted this question Tangent space of $G \times_H M$ in MSE, but didn't get any answer. My question is the following:

Let $G$ be a Lie group and let $H$ be a Lie subgroup of $H$. Let $M$ be a smooth manifold on which $H$ acts from the left.

Let's consider the action of $H$ on $G \times M$: $$h((g,m)):= (gh,h^{-1}m), \quad h \in H , g \in G , m \in M, $$ and define the manifold $Z$ to be the quotient $G \times_H M$.

If we fix $(g,m) \in G \times M$, what is the induced equivalence relation on $T_gG \times T_mM$?

(My background is not so good in differential geometry, so please be patient with me.)