The fusion of $\alpha$ and ${}^8\mathrm{Be}$ to produce a ${}^{12}\mathrm{C}$ nucleus is a crucial process in nucleosynthesis. In the laboratory, this process can only be studied theoretically because a ${}^8\mathrm{Be}$ target or projectile cannot be prepared experimentally. We use the potential scattering theory in the coupled-channel formalism to study such a process in terms of the collision between the $\alpha$ particle on a deformed ${}^8\mathrm{Be}$ nucleus, both on resonance and off resonance in the Hoyle resonance and associated resonances region. The experimental ${}^{12}\mathrm{C}$ energy levels and widths constrain the nuclear potential to suggest the need to include a parity-dependent surface potential component that is more attractive for even-$L$ positive-parity partial waves than for odd-$L$ negative-parity partial waves. As a consequence, the radial dependence of the total potentials for the set of {$0^{+}$, $2^{+}$, $4^{+}$} resonances of ${}^{12}\mathrm{C}$ exhibit a double-hump behavior, possessing two local energy minima and a doublet of each of the ${}^{12}\mathrm{C}$ {$0^{+}$, $2^{+}$, $4^{+}$} resonances in the Hoyle and associated resonances region. We examine the approximate agreement of the theoretical results with experiment and suggest the search for the as-yet unobserved lower-energy $2_2^{+}$ and $4{}_1^{+}$ resonances to test the double-hump potential description. In addition, for practical astrophysical applications, we evaluate and estimate the astrophysical $S\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ factor for the $\alpha +{}^8\mathrm{Be}$ $\to$ ${}^{12}\mathrm{C}\left(0^{+*}\right)$ $\to$ ${}^{12}\mathrm{C}\left(2_1^{+}\right)$ $+$ $\gamma$ reaction for $E_{\mathrm{c}.\mathrm{m}.}$ $<$ 1.0 MeV.

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• Formation & evolution of stars & galaxies
• Nuclear astrophysics
• Nuclear reactions
• Nucleosynthesis in explosive environments
• 6 ≤ A ≤ 19