  We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation
theorems from the setting of free groups to that of free products. As a result,
we obtain a fixed point theorem for finite groups of outer automorphisms acting
on the relative free splitting complex of Handel--Mosher and on the outer space
of a free product of Guirardel--Levitt, as well as a relative version of the
Nielsen realisation theorem, which in the case of free groups answers a
question of Karen Vogtmann. We also prove Nielsen realisation for limit groups,
and as a byproduct obtain a new proof that limit groups are CAT($0$). The
proofs rely on a new version of Stallings' theorem on groups with at least two
ends, in which some control over the behaviour of virtual free factors is
gained.
