  We determine all natural operations and their relations on the homotopy
groups of spectral partition Lie algebras, which coincide with
$\mathbb{F}_p$-linear topological Andr\'{e}-Quillen cohomology operations at
any prime. We construct unary operations and a shifted restricted Lie algebra
structure on the homotopy groups of spectral partition Lie algebras. Then we
prove a composition law for the unary operations, as well as a compatibility
condition between unary operations and the shifted Lie bracket with restriction
up to a unit for the restriction. Comparing with Brantner-Mathew's result on
the ranks of the homotopy groups of free spectral partition Lie algebras, we
deduce that these generate all natural operations and obtain the target
category, thereby also recovering unpublished results of Kriz and
Basterra-Mandell on $\mathbb{F}_p$-linear TAQ cohomology operations. As a
corollary, we determine the structure of natural operations on mod $p$
$\mathbb{S}$-linear TAQ cohomology.
