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Abstract
In 2009, J. Wood [43] proved that Frobenius bimodules have the extension property for symmetrized weight compositions. More generally, in [11], it is shown that having a cyclic socle is suf{uFB01}cient for satisfying the property, while the necessity remained an open question. Here, landing in Midway, a partial converse is proved. For a signi{uFB01}cant class of {uFB01}nite module alphabets, the cyclic socle is shown necessary to satisfy the ex- tension property. The idea is bridging to the case of Hamming weight through a new weight function