High-energy-resolution ${}^{56}$Fe(${}^{3}\mathrm{He},t$)${}^{56}$Co reaction measurements were performed at forward angles including ${0}^{\ensuremath{\circ}}$. From the spectra obtained, the splitting of the Fermi transition strength and thus the isospin mixing between the ${J}^{\ensuremath{\pi}}={0}^{+}$ isobaric analog state (IAS) at 3.60 MeV and a neighboring ${0}^{+}$ state at 3.53 MeV have been investigated. In order to distinguish between states excited in Fermi and Gamow-Teller (GT) transitions, the measurements were performed at two ${}^{3}$He beam energies of 140 and 100 MeV/nucleon. Owing to the different incident energy dependencies of the $\ensuremath{\sigma}\ensuremath{\tau}$- and $\ensuremath{\tau}$-type effective interaction strengths, it is expected that the Fermi transitions are stronger relative to GT transitions at lower beam energies. Therefore the excitation of a state by the Fermi transition could be identified by the $\ensuremath{\Delta}L=0$ angular distribution and the ratio of transition strengths at these two incident energies. In the energy region around the IAS at 3.60 MeV, it was found that the state at 3.53 MeV is also excited by the Fermi transition and thus it has ${J}^{\ensuremath{\pi}}$ values of ${0}^{+}$. A value of 32.3(5) keV could be deduced for the off-diagonal matrix element of the Hamiltonian that causes the isospin mixing. A corresponding isospin impurity of $28\ifmmode\pm\else\textpm\fi{}1$% was obtained.