Lifting wreath product extensions

Let $G$ and $H$ be finite groups and let $K$ be a hilbertian field. We show that if $G$ has a generic extension over $K$ and $H$ satisfies the arithmetic lifting property over $K$, then the wreath product $G\wr H$ of $G$ and $H$ also satisfies the arithmetic lifting property over $K$. Moreover, if the orders of $H$ and $G$ are relatively prime and $G$ is abelian, then any extension of $G$ by $H$(which is necessarily a semidirect product) has the arithmetic lifting property.