Nuclear deexcitation lines are regularly observed in the $\gamma$-ray emission spectra of strong solar flares. The most prominent lines are produced by interactions of protons and $\alpha$ 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 $\gamma$-ray line emission in these interactions for a wide particle energy range. To this purpose, we measured the $\gamma$-ray emission in interactions of $\alpha$-particle beams of $E_{\alpha }=50\text{–}90$ 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 $\gamma$ 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 $\gamma$-ray emission from the targets from other $\gamma$ rays and neutron-induced background. $\gamma$-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 $E_{\alpha }=90$ MeV that can be directly used for astrophysical applications like solar flares. Comparison of the results with a cross-section compilation for strong $\gamma$-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.