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In this paper, which is a follow-up to [A. Borobia, R. Canogar, F. DeTer\'an, Mediterr. J. Math. 18, 40 (2021)], we provide a necessary andsufficient condition for the matrix equation $X^\top AX=B$ to be consistentwhen $B$ is symmetric. The condition depends on the canonical form forcongruence of the matrix $A$, and is proved to be necessary for all matrices$A$, and sufficient for most of them. This result improves the main one in theprevious paper, since the condition is stronger than the one in that reference,and the sufficiency is guaranteed for a larger set of matrices (namely, thosewhose canonical form for congruence, CFC($A$), includes skew-symmetric blocks).

On the consistency of the matrix equation $X^\top A X=B$ when $B$ is symmetric: the case where CFC($A$) includes skew-symmetric blocks