If H/K is normal in G/K, then is H normal in G? Let $G$ be a group and $H, K$ are two subgroups of it s.t. $K$ is normal in $G$. Now if $H/K$ is a normal subgroup in $G/K$, then can we say $H$ is a normal subgroup of $G$?

Yes, this is the Lattice or fourth isomorphism theorem. $H$ is normal in $G$ if and only if $H/K$ is normal in $G/K$ for $K$ a normal subgroup of $G$.
