We compute the two-loop effects induced by an anomalous Higgs trilinear self-coupling in the partial decay width $$\Gamma (h \rightarrow \gamma Z)$$ $\Gamma (h\to \gamma Z)$ . The computation is performed using the anomalous coupling approach, working in the unitary gauge, and in a theory in which the anomalous coupling is generated via the addition to the scalar potential part of the Standard Model Lagrangian of an (in)finite tower of $$(\Phi ^\dagger \Phi )^n$$ ${\left({\Phi }^{†}\Phi \right)}^n$ terms. The former computation is automatically finite while the latter requires the renormalization of the lowest order contribution. We discuss the renormalization conditions that should be employed in order to obtain the same result in the two approaches. We find that the $$h \rightarrow \gamma Z$$ $h\to \gamma Z$ process is one of the most sensitive mode to an anomalous trilinear Higgs self-coupling. As a by-product of this work we confirm one of two different results present in the literature concerning the contribution of an anomalous Higgs trilinear coupling in the $$h \rightarrow \gamma \gamma $$ $h\to \gamma \gamma$ decay.