  We present an advanced method to study spin fluctuations in superconductors
quantitatively, and entirely from first principles. This method can be
generally applied to materials where electron-phonon coupling and spin
fluctuations coexist. We employ it here to examine the recently synthesized
superconductor iron tetraboride (FeB$_4$) with experimental $T_{\mathrm{c}}\sim
2.4$ K [H. Gou \textit{et al.}, Phys. Rev. Lett. \textbf{111}, 157002 (2013)].
We prove that FeB$_4$ is particularly prone to ferromagnetic spin fluctuations
due to the presence of iron, resulting in a large Stoner interaction strength,
$I=1.5$ eV, as calculated from first principles. The other important factor is
its Fermi surface that consists of three separate sheets, among which two
nested ellipsoids. The resulting susceptibility has a ferromagnetic peak around
$\textbf{q}=0$, from which we calculated the repulsive interaction between
Cooper pair electrons using the random phase approximation. Subsequently, we
combined the electron-phonon interaction calculated from first principles with
the spin fluctuation interaction in fully anisotropic Eliashberg theory
calculations. We show that the resulting superconducting gap spectrum is
conventional, yet very strongly depleted due to coupling to the spin
fluctuations. The critical temperature decreases from $T_{\mathrm{c}}= 41$ K,
if they are not taken into account, to $T_{\mathrm{c}}= 1.7$ K, in good
agreement with the experimental value.
