We evaluate partition functions ${Z}_{I}$ in topologically nontrivial (instanton) gauge sectors in the bosonized version of the Schwinger model and in a gauged WZNW model corresponding to two-dimensional QCD (QC${\mathrm{D}}_{2}$) with adjoint fermions. We show that the bosonized model is equivalent to the fermion model only if a particular form of the WZNW action with a gauge-invariant integrand is chosen. For the exact correspondence, it is necessary to integrate over the ways the gauge group $\frac{\mathrm{SU}(N)}{{Z}_{N}}$ is embedded into the full $\mathrm{O}({N}^{2}\ensuremath{-}1)$ group for the bosonized matter field. For even $N$, one should also take into account the contributions of both disconnected components in $\mathrm{O}({N}^{2}\ensuremath{-}1)$. In that case, ${Z}_{I}\ensuremath{\propto}{m}^{{n}_{0}}$ for small fermion masses where $2{n}_{0}$ coincides with the number of fermion zero modes in a particular instanton background. The Taylor expansion of $\frac{{Z}_{I}}{{m}^{{n}_{0}}}$ in mass involves only even powers of $m$, as it should. The physics of adjoint QC${\mathrm{D}}_{2}$ is discussed. We argue that, for odd $N$, the discrete chiral symmetry ${Z}_{2}\ensuremath{\bigotimes}{Z}_{2}$ present in the action is broken spontaneously down to ${Z}_{2}$ and the fermion condensate ${〈\overline{\ensuremath{\lambda}}\ensuremath{\lambda}〉}_{0}$ is formed. The system undergoes a first order phase transition at ${T}_{c}=0$ so that the condensate is zero at an arbitrary small temperature. It is not yet quite clear what happens for even $N>~4$.