  Let $M$ be an orientable, irreducible $3$-manifold admitting a weakly
reducible genus three Heegaard splitting as a minimal genus Heegaard splitting.
In this article, we prove that if $[f]$, $[g]\in Mod(M)$ give the same
correspondence between two isotopy classes of generalized Heegaard splittings
consisting of two Heegaard splittings of genus two, say
$[\mathbf{H}]\to[\mathbf{H}']$, then there exists a representative $h$ of the
difference $[h]=[g]\cdot[f]^{-1}$ such that (i) $h$ preserves a suitably chosen
embedding of the Heegaard surface $F'$ obtained by amalgamation from
$\mathbf{H}'$ which is a representative of $[\mathbf{H}']$ and (ii) $h$ sends a
uniquely determined weak reducing pair $(V',W')$ of $F'$ into itself up to
isotopy. Moreover, for every orientation-preserving automorphism $\tilde{h}$
satisfying the previous conditions (i) and (ii), there exist two elements of
$Mod(M)$ giving correspondence $[\mathbf{H}]\to[\mathbf{H}']$ such that
$\tilde{h}$ belongs to the isotopy class of the difference between them.
