  In this paper we compute common neighbourhood (abbreviated as CN) spectrum
and energy of commuting conjugacy class graph of several families of finite
non-abelian groups. As a consequence of our results we show that the commuting
conjugacy class graphs of the groups $D_{2n}$, $T_{4n}$, $SD_{8n}$,
$U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ and some families of groups whose
central quotient is isomorphic to $D_{2n}$ or $Z_p \times Z_p$, for some prime
$p$, are CN-integral but not CN-hyperenergetic.
