  We study the categorical-algebraic condition that internal actions are weakly
representable (WRA) in the context of varieties of (non-associative) algebras
over a field.
  Our first aim is to give a complete characterization of action accessible,
operadic quadratic varieties of non-associative algebras which satisfy an
identity of degree two and to study the representability of actions for them.
Here we prove that the varieties of two-step nilpotent (anti-)commutative
algebras and that of commutative associative algebras are weakly action
representable, and we explain that the condition (WRA) is closely connected to
the existence of a so-called amalgam.
  Our second aim is to work towards the construction, still within the context
of algebras over a field, of a weakly representing object $E(X)$ for the
actions on (or split extensions of) an object $X$. We actually obtain a partial
algebra $E(X)$, which we call external weak actor of $X$, together with a
monomorphism of functors ${\operatorname{SplExt}(-,X) \rightarrowtail
\operatorname{Hom}(U(-),E(X))}$, which we study in detail in the case of
quadratic varieties. Furthermore, the relations between the construction of the
universal strict general actor $\operatorname{USGA}(X)$ and that of $E(X)$ are
described in detail. We end with some open questions.
