  In this paper, we characterize the finite groups $G$ of even order with the
property that for any involution $x$ and element $y$ of $G$, $\langle x, y
\rangle$ is isomorphic to one of the following groups: $\mathbb{Z}_2,$
$\mathbb{Z}_2^2$, $\mathbb{Z}_4$, $\mathbb{Z}_6$, $\mathbb{Z}_2 \times
\mathbb{Z}_4$, $\mathbb{Z}_2 \times\mathbb{Z}_6$ and $A_4$. As a result, a
characterization will be obtained for the finite groups all of whose Cayley
graphs of degree $3$ have integral spectrum.
