  We prove that if a surface group embeds as a normal subgroup in a K\"ahler
group and the conjugation action of the K\"ahler group on the surface group
preserves the conjugacy class of a non-trivial element, then the K\"ahler group
is virtually given by a direct product, where one factor is a surface group.
Moreover we prove that if a one-ended hyperbolic group with infinite outer
automorphism group embeds as a normal subgroup in a K\"ahler group then it is
virtually a surface group. More generally we give restrictions on normal
subgroups of K\"ahler groups which are amalgamated products or HNN extensions.
