  In 2009, J. Wood [15] proved that Frobenius bimodules have the extension
property for symmetrized weight compositions. Later, in [9], it was proved that
having a cyclic socle is sufficient for satisfying the property, while the
necessity remained an open question. Here, landing in Midway, the necessity is
proved, a module alphabet RA has the extension property for symmetrized weight
compositions built on AutR(A) is necessarily having a cyclic socle.
