  Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined
over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph
which has a vertex for each element of $\mathcal{E}$ and an edge for each
$\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in
$\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree of
the isogeny recorded as a label of the edge. The isogeny-torsion graph
associated to $\mathcal{E}$ is the isogeny graph associated to $\mathcal{E}$
where, in addition, we label each vertex with the abstract group structure of
the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. The
main result of the article is a classification of the 2-adic Galois image at
each vertex of the isogeny-torsion graphs whose associated $\mathbb{Q}$-isogeny
class consists of elliptic curves over $\mathbb{Q}$ with complex
multiplication.
