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In this paper, we are concerned with backward solvabilities of heatequations, in an abstract framework. We show that semigroups $T_t$ in Banachspaces $X$, generated by heat operators, are extendable to groups in anextended space $E$, which is obtained by considering a sequence of wider Banachspaces containing $X$, i.e. $X$$/subset$$X_t$$/subset$$X_s$... $(t<s)$, underthe following two conditions. One is the density assumption on a subset $D$ of$X$, the set of initial values $x$ from which $T_{-t}x$ exists for all $t>0$.Another is the backward uniqueness of the semigroup $T_t$. For example, weprove the holomorphic semigroup satisfies the above conditions, and thus isextendable to a group in a larger functional space $E$. We also studiedstructual properties of the extended space $E$.

Extended groups of semigroups and backward problems of heat equations