  It is a longstanding conjecture that for a finite group $G$, the exponent of
the second homology group $H_2(G, \mathbb{Z})$ divides the exponent of $G$. In
this paper, we prove this conjecture for $p$-groups of class at most $p$,
finite nilpotent groups of odd exponent and of nilpotency class 5, $p$-central
metabelian $p$-groups, and groups considered by L. E . Wilson in \cite{LEW}.
Moreover, we improve several bounds given by various authors. We achieve most
of our results using an induction argument.
