Is it always true, that if $G$ is a group and $A$ is a subgroup of $G$ then $|G:A|<\infty\Rightarrow|G:A^x|<\infty$

Question

Is it always true, that if $G$ is a group and $A$ is a subgroup of $G$ then $$|G:A|<\infty\Rightarrow|G:A^x|<\infty$$ where $x\in G$.

I did as follows: let $|G:A|=n\Rightarrow G=g_1A\sqcup\cdots\sqcup g_nA$ then $$G=G^x=(g_1A)^x\sqcup\cdots\sqcup(g_nA)^x=$$ $$=g_1^xA^x\cup\cdots\cup g_n^xA^x$$ $\Rightarrow |G:A^x|\le|G:A|\lt \infty$.

But I'm not sure if I solved the problem correctly. Please, help me.