The Lyman-$alpha$ forest is a powerful probe for cosmology, but it is also strongly impacted by galaxy evolution and baryonic processes such as Active Galactic Nuclei (AGN) feedback, which can redistribute mass and energy on large scales. We constrain the signatures of AGN feedback on the 1D power spectrum of the Lyman-$alpha$ forest using a series of eight hydro-cosmological simulations performed with the Adaptative Mesh Refinement code RAMSES. This series starts from the Horizon-AGN simulation and varies the sub-grid parameters for AGN feeding, feedback and stochasticity. These simulations cover the whole plausible range of feedback and feeding parameters according to the resulting galaxy properties. AGNs globally suppress the Lyman-$alpha$ power at all scales. On large scales, the energy injection and ionization dominate over the supply of gas mass from AGN-driven galactic winds, thus suppressing power. On small scales, faster cooling of denser gas mitigates the suppression. This effect increases with decreasing redshift. We provide lower and upper limits of this signature at nine redshifts between $z=4.25$ and $z=2.0$, making it possible to account for it at post-processing stage in future work given that running simulations without AGN feedback can save considerable amounts of computing resources. Ignoring AGN feedback in cosmological inference analyses leads to strong biases with 2% shift on $sigma_8$ and 1% shift on $n_s$, which represents twice the standards deviation of the current constraints on $n_s$.
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