If $A$ is just one point $p$ then you can define $s$ by the required equivarianace property on orbit of $p$ and then you can extend it to whole $M$ since you said that $M/G$ is a manifold.

On the other hand, if $g \in G$ is sufficiently close to identity, then $g\cdot p$ will be close to $p$ and your $s_1$ must itself satisfy some equivariance with respect to some small neighborhood of identity element of $G$. So if you want bigger $A$ you need some assumptions on $s_1$.

If $A$ is just one point $p$ then you can define $s$ by the required equivarianace property on orbit of $p$ and then you can extend it to whole $M$ since you said that $M/G$ is a manifold.

On the other hand, if $g \in G$ is sufficiently close to identity, then $g\cdot p$ will be close to $p$ and your $s_1$ must itself satisfy equivariance with respect to some small neighborhood of identity element of $G$. So if you want bigger $A$ you need some assumptions on $s_1$.