In conditions of $4\pi$-geometry, the production of ${\Delta }^0$ and ${\Delta }^{++}$ resonances in collisions of ${}^4\mathrm{He}$ nuclei with carbon nuclei at 4.2 GeV/$c$ per nucleon was investigated for the first time. The masses and widths of resonances were obtained based on an analysis of the experimental and background invariant mass distributions of $p{\pi }^{\pm }$ pairs. The contributions to the pion production from decays of $\Delta$ resonances, as well as from direct pion creation, and the relative number of nucleons excited to ${\Delta }^0$ at freeze-out were estimated. The results obtained were compared to the corresponding data for C+C collisions at the same initial momentum 4.2 GeV/$c$ per nucleon.