  In \cite{1}, a class of gyrogroups $(G,\odot)$ has been associated to certain
groups $(G,\cdot)$. We give a necessary and sufficient condition for
$(G,\odot)$ to be gyrocommutative. We also prove that under a suitable
assumption two finite groups central by a $2$-Engel group are isomorphic if and
only if their associated gyrogroups are isomorphic.
