Nil_2(Grp) is weakly action representable with an abelian weak representing object but neither action representable nor locally algebraically cartesian closed.
Gray,Algebraic exponentiation for categories of Lie algebras, Journal of Pure and Applied Algebra216(2012), no
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Weak action representability of 2-nilpotent groups
Nil_2(Grp) is weakly action representable with an abelian weak representing object but neither action representable nor locally algebraically cartesian closed.