I am reading Duistermaat's 1973 paper on relating the convexity of the image of a moment map to the image of the fixed points of an antisymplectic involution. In that paper, the following comment is made:
Let $M$ be a smooth manifold endowed with a smooth action of a compact group $G$. If $\beta$ is a $G$-invariant Riemannian metric (always possible since $G$ is compact) and $m \in M$ is a fixed point of the $G$-action, then the $\beta$-exponential map at $m$ intertwines the linear action of $G$ on $T_mM$ and the local action of $G$ around $m$.
My interpretation of this statement is as follows: If $A_g: M \to M, m\mapsto g\cdot m$ denotes the action of $g \in G$ and $\exp_m: T_m M \to M$ is the $\beta$-exponential map, then $$ \exp_m \circ d_m A_g = A_g\circ \exp_m.$$
Now I believe this is true if $G=M$, since in that case it is known that the $\beta$-Riemannian and Lie exponential coincide (though I would love a reference for this) and this should boil down to a statement regarding the the functorial nature of the exponential, but it is not clear to me why this is true in the more general case.