 Solution

Conjugation ${\alpha }_g(x)=gx{g}^{-1}$ is an automorphism of $G$ sending $A$ to $gA{g}^{-1}$. It induces the bijectionbetween their left coset sets; conjugation by $g^{-1}$ gives the inverse. ThereforeThis holds whether or not $A$ is normal, and whether the index is finite or infinite.