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Let $M$ be an orientable, irreducible $3$-manifold admitting a weaklyreducible genus three Heegaard splitting as a minimal genus Heegaard splitting.In this article, we prove that if $[f]$, $[g]\in Mod(M)$ give the samecorrespondence between two isotopy classes of generalized Heegaard splittingsconsisting of two Heegaard splittings of genus two, say$[\mathbf{H}]\to[\mathbf{H}']$, then there exists a representative $h$ of thedifference $[h]=[g]\cdot[f]^{-1}$ such that (i) $h$ preserves a suitably chosenembedding of the Heegaard surface $F'$ obtained by amalgamation from$\mathbf{H}'$ which is a representative of $[\mathbf{H}']$ and (ii) $h$ sends auniquely determined weak reducing pair $(V',W')$ of $F'$ into itself up toisotopy. 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.

On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three III - Generalized Heegaard splittings and mapping classes