  Renormalization group methods are an essential ingredient in the study of
nonperturbative problems of quantum field theory. This paper deal with the
symmetry constraints on the renormalization group flow for quartic melonic
tensorial group field theories. Using the unitary invariance of the
interactions, we provide a set of Ward-Takahashi identities which leads to
relations between correlation functions. There are numerous reasons to consider
such Ward identities in the functional renormalization group. Their
compatibility along the flow provides a non-trivial constraint on the
reliability of the approximation schemes used in the non-perturbative regime,
especially on the truncation and the choice of the regulator. We establish the
so called structure equations in the melonic sector and in the symmetric phase.
As an example we consider the $T^4_5$ TGFT model without gauge constraint. The
Wetterich flow equation is given and the way to improve the truncation on the
effective action is also scrutinized.
