  We undertake a systematic study of the so-called $2$-adic ring $C^*$-algebra
$\mathcal{Q}_2$. This is the universal $C^*$-algebra generated by a unitary $U$
and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$.
Notably, it contains a copy of the Cuntz algebra $\mathcal{O}_2=C^*(S_1, S_2)$
through the injective homomorphism mapping $S_1$ to $US_2$. Among the main
results, the relative commutant $C^*(S_2)'\cap \mathcal{Q}_2$ is shown to be
trivial. This in turn leads to a rigidity property enjoyed by the inclusion
$\mathcal{O}_2\subset\mathcal{Q}_2$, namely the endomorphisms of
$\mathcal{Q}_2$ that restrict to the identity on $\mathcal{O}_2$ are actually
the identity on the whole $\mathcal{Q}_2$. Moreover, there is no conditional
expectation from $\mathcal{Q}_2$ onto $\mathcal{O}_2$. As for the inner
structure of $\mathcal{Q}_2$, the diagonal subalgebra $\mathcal{D}_2$ and
$C^*(U)$ are both proved to be maximal abelian in $\mathcal{Q}_2$. The
maximality of the latter allows a thorough investigation of several classes of
endomorphisms and automorphisms of $\mathcal{Q}_2$. In particular, the
semigroup of the endomorphisms fixing $U$ turns out to be a maximal abelian
subgroup of ${\rm Aut}(\mathcal{Q}_2)$ topologically isomorphic with
$C(\mathbb{T},\mathbb{T})$. Finally, it is shown by an explicit construction
that ${\rm Out}(\mathcal{Q}_2)$ is uncountable and non-abelian.
