I asked this a week ago at MSE, but nobody replied. Could anybody enlighten me if the following is true?

Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces over $k$). Let $$ \varphi:F\to G $$ be a morphism of Hopf algebras (i.e. $\varphi$ preserves multiplication, comultiplication, unit, counit and antipode), and $$ H\subseteq G $$ a Hopf subalgebra in $G$ (i.e. $H$ is a Hopf algebra with respect to the multiplication, comultiplication, unit, counit and antipode induced from $G$).

Question:

Is it true that the preimage of $H$, $$ \varphi^{-1}(H)=\{x\in F:\ \varphi(x)\in H\} $$ is a Hopf subalgebra in $F$?