Summary: Let \((L, \cdot )\) be any loop and let \(A(L)\) be a group of automorphisms of \((L, \cdot )\) such that \(\alpha\) and \(\phi\) are elements of \(A(L)\). It is shown that, for all \(x, y, z \in L\), the \(A(L)\)-holomorph \((H, \circ) = H(L)\) of \((L, \cdot )\) is an Osborn loop if and only if \(x \alpha(yz \cdot x \phi^{-1} ) = x \alpha (yx^\lambda \cdot x) \cdot zx \phi^{-1} \). Furthermore, it is shown that for all \(x \in L\), \(H(L)\) is an Osborn loop if and only if \((L, \cdot )\) is an Osborn loop, \((x \alpha \cdot x^\rho )x = x \alpha\), \(x(x^\lambda \cdot x \phi^{-1} ) = x \phi^{-1}\) and every pair of automorphisms in \(A(L)\) is nuclear (i.e. \(x \alpha \cdot x^\rho\), \(x^\lambda \cdot x \phi \in N(L, \cdot)\)). It is shown that if \(H(L)\) is an Osborn loop, then \(A(L, \cdot ) = \mathcal{P} (L, \cdot) \cap \Lambda (L, \cdot) \cap \Phi (L, \cdot) \cap \Psi (L, \cdot )\) and for any \(\alpha \in A(L)\), \(\alpha = L_{e\pi } = R_{e\varrho }^{ - 1}\) for some \(\pi \in \Phi (L, \cdot )\) and some \(\varrho \in \Psi (L, \cdot )\). Some commutative diagrams are deduced by considering isomorphisms among the various groups of regular bijections (whose intersection is \(A(L)\)) and the nucleus of \((L, \cdot )\).