We show that every action of a finite dihedral group on a closed orientable surface
F
\mathcal F
extends to a 3-dimensional handlebody
V
\mathcal V
, with
∂
V
=
F
\partial \mathcal V=\mathcal F
. In the case of a finite abelian group
G
G
, we give necessary and sufficient conditions for a
G
G
-action on a surface to extend to a compact
3
3
-manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order
84
(
g
−
1
)
84(g-1)
of
P
S
L
(
2
,
27
)
PSL(2,27)
on a surface
F
\mathcal F
of genus
g
=
118
g=118
does not extend to any compact 3-manifold
M
M
with
∂
M
=
F
\partial M=\mathcal F
, thus resolving the only case of Hurwitz actions of type
P
S
L
(
2
,
q
)
PSL(2,q)
of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151).