<p>We introduce a notion of action for unitary magmas and use it to classify a broad class of split extensions, called retraction points. Our approach is based on previous work on split extensions and semidirect products of unitary magmas, extending it to the more general setting in which the middle object of a split extension need not be in bijection with the Cartesian product of the kernel and cokernel objects. The construction is based on the notion of semibiproduct rather than the classical semidirect product of groups. Although this phenomenon does not occur in groups or, more generally, in associative semiabelian varieties, it appears in monoids through weakly Schreier extensions and in certain non-associative structures such as semi-left-loops. These results provide a broader framework for the study of split extensions of unitary magmas and related categorical structures.</p>