Summary: In 1996, Friedgut and Kalai made the Fourier entropy-influence conjecture: For every Boolean function $f : \{-1, 1\}^{n} \longrightarrow \{-1, 1\}$ it holds that $H[{\hat{f}}^{2}] \leq C \cdot I[f]$, where $H[{\hat{f}}^{2}]$ is the spectral entropy of $f$, $I[f]$ is the total influence of $f$, and $C$ is a universal constant. In this work we verify the conjecture for symmetric functions. More generally, we verify it for functions with symmetry group $S_{n_1} \times \cdots \times S_{n_d}$ where $d$ is constant. We also verify the conjecture for functions computable by read-once decision trees.