  In this paper we study monodromy operators on moduli spaces $M_v(S,H)$ of
sheaves on K3 surfaces with non-primitive Mukai vectors $v$. If we write
$v=mw$, with $m>1$ and $w$ primitive, then our main result is that the
inclusion $M_w(S,H)\to M_v(S,H)$ as the most singular locus induces an
isomorphism between the monodromy groups of these symplectic varieties,
allowing us to extend to the non-primitive case a result of Markman.
