I'm learning about the Sylow theorem and the p-Sylow subgroups. In certain applications of the Sylow Theorem one considers the $p$-Sylow subgroups of the quotient group $G/H$ to conclude a statement about a $p$-Sylow subgroup in the group $G$.
What is the general relation between them?
If $G$ is a group such that ${\rm card}(G) = p^s \cdot m$ and such that $\gcd(p,m) = 1$. I think that if $P$ is a unique $p$-Sylow subgroup of $G/H$ with order $p^s$ then there will be a unique $P'$ such that $P'$ is a $p$-Sylowsubgroup but in $G$ with the same order. Is that true and/or are there any other known relations?
Thanks for your help!