  The notion of linear Hahn-Banach extension operator was first studied in
detail by Heinrich and Mankiewicz (1982). Previously, J. Lindenstrauss (1966)
studied similar versions of this notion in the context of non separable
reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the
existence of linear Hahn-Banach extension operators via interspersing subspaces
in a purely Banach space theoretic set up. In this paper, we study similar
questions in the context of Banach modules and module homomorphisms, in
particular, Banach algebras of operators on Banach spaces. Based on Dales,
Kania, Kochanek, Kozmider and Laustsen(2013), and also Kania and Laustsen
(2017), we give complete answers for reflexive Banach spaces and the
non-reflexive space constructed by Kania and Laustsen from the celebrated
Argyros-Haydon's space with few operators.
