  In this paper we continue the study started recently in \cite{ABMbp} by
describing and classifying all Hopf algebras $E$ that factorize through two
Sweedler's Hopf algebras. Equivalently, we classify all bicrossed products $H_4
\bowtie H_4$. There are three steps in our approach. First, we explicitly
describe the set of all matched pairs $(H_4, H_4, \triangleright,
\triangleleft)$ by proving that, with the exception of the trivial pair, this
set is parameterized by the ground field $k$. Then, for any $\lambda \in k$, we
describe by generators and relations the associated bicrossed product,
\mathcal{H}_{16, \, \lambda}$. This is a 16-dimensional, pointed, unimodular
and non-semisimple Hopf algebra. A Hopf algebra $E$ factorizes through $H_4$
and $H_4$ if and only if $ E \cong H_4 \ot H_4$ or $E \cong \mathcal{H}_{16,\,
\lambda}$. In the last step we classify these quantum groups by proving that
there are only three isomorphism classes represented by: $H_4 \ot H_4$,
$\mathcal{H}_{16, \, 0}$ and $\mathcal{H}_{16, \, 1} \cong D(H_4)$, the
Drinfel'd double of $H_4$. The automorphism group of these objects is also
computed: in particular, we prove that $\Aut_{\rm Hopf}\big(D(H_4)\big)$ is
isomorphic to a semidirect product of groups, $k^{\times} \rtimes
\mathbb{Z}_2$.
