On some invariants under the action of an extension of $GA(n,2)$ on the set of Boolean functions

Let $G$ be the extension of a general affine group by the group of affine functions. We study the action of $G$ on the set of Boolean functions. The action consists in nondegenerate affine transformations of variables and addition of affine Boolean functions. We introduce and examine some parameters of Boolean functions which are invariant with respect to the action of $G$. These are the amplitude (which is closely related to the nonlinearity), the dimension of a function, and some others. The invariants, together with some additionally proposed notions, could be used to obtain new bounds on cryptographic parameters of Boolean functions, including the maximum nonlinearity of functions in an odd number of variables.