Nuclear deexcitation lines are regularly observed in the $\ensuremath{\gamma}$-ray emission spectra of strong solar flares. The most prominent lines are produced by interactions of protons and $\ensuremath{\alpha}$ particles, accelerated up to hundreds of MeV, with abundant nuclei of the solar atmosphere. Analysis and interpretation of these lines, which carry valuable information on the solar flare properties, need cross-section data for the $\ensuremath{\gamma}$-ray line emission in these interactions for a wide particle energy range. To this purpose, we measured the $\ensuremath{\gamma}$-ray emission in interactions of $\ensuremath{\alpha}$-particle beams of ${E}_{\ensuremath{\alpha}}=50\text{--}90$ MeV with target foils of C, Mg, Si, and Fe at the center for proton therapy of the Helmholtz-Zentrum Berlin. Setups of three high-purity Ge detectors and one $\mathrm{La}{\mathrm{Br}}_{3}$:Ce detector have been employed to detect the $\ensuremath{\gamma}$ rays in two experiment campaigns. Relatively large distances of the detectors from the target and pulsed beams with sub-ns-wide pulses allowed the separation of beam-induced prompt $\ensuremath{\gamma}$-ray emission from the targets from other $\ensuremath{\gamma}$ rays and neutron-induced background. $\ensuremath{\gamma}$-ray production cross sections for about 60 deexcitation lines from excited target nuclei or reaction products have been determined. For the strongest deexcitation lines from the major target isotopes, $^{12}\mathrm{C}$, $^{24}\mathrm{Mg}$, $^{28}\mathrm{Si}$, $^{56}\mathrm{Fe}$, there are now measured cross-section data from reaction threshold to ${E}_{\ensuremath{\alpha}}=90$ MeV that can be directly used for astrophysical applications like solar flares. Comparison of the results with a cross-section compilation for strong $\ensuremath{\gamma}$-ray lines in solar flare emissions and the predictions of the talys nuclear reaction code were done. They underline the importance of cross-section determinations at accelerator laboratories for the establishment of an accurate cross-section data base in a wide projectile energy range.