In this paper we study monodromy operators on moduli spaces
M
v
(
S
,
H
)
M_v(S,H)
of sheaves on K3 surfaces with non-primitive Mukai vectors
v
v
. If we write
v
=
m
w
v=mw
, with
m
>
1
m>1
and
w
w
primitive, then our main result is that the inclusion
M
w
(
S
,
H
)
→
M
v
(
S
,
H
)
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.