  The main purpose of this paper is to show that the converse of the known
implication weakly action representable implies action accessible is false. In
particular we show that both action accessibility, as well as the (at least
formally stronger) condition requiring the existence of all normalizers do not
imply weakly-action-representability even for varieties. In addition we show
that in contrast to both action accessibility and the condition requiring the
existence of all normalizers, weakly-action representability is not necessarily
inherited by Birkoff subcategories.
