Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal

Statement with symbols
If $$H$$ is an automorph-join-closed subnormal subgroup of $$K$$ and $$K$$ is a normal subgroup of $$G$$, then $$H$$ is a conjugate-join-closed subnormal subgroup of $$G$$.

Related facts

 * Left residual of conjugate-join-closed subnormal by normal equals automorph-join-closed subnormal
 * Finite-automorph-join-closed subnormal of normal implies finite-conjugate-join-closed subnormal
 * Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormal
 * 3-subnormal implies finite-conjugate-join-closed subnormal