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The germs of maps $(k^n,o)\stackrel{f}{\to}(k^p,o)$ are traditionally studiedup to the right (R), left-right (A) or contact (K) equivalence. Variousquestions about the group-orbits $G f$ are transformed to the tangent spaces,$T_G f$. Classically this passage was done by vector fields integration, henceit was bound to the (real/complex) analytic or $C^r$-category.  The characteristic-free approach to the groups $R,K$ has been developedduring the last decade. The results were still restricted. E.g. one could notaddress the (essentially more complicated) $A$-equivalence. Moreover, thecharacteristic-free criteria were weaker than those in characteristic zero,because of the (inevitable) pathologies of positive characteristic.  In this paper we close these omissions.  We establish the general (characteristic-free) passage $T_G f\rightsquigarrowG f$ for the groups $R$, $K,$ $A$. Submodules of $T_G f$ ensure submodules of$G f$. For the $A$-equivalence this extends (and strengthens) various classicalresults by Mather, Gaffney, du Plessis, and others.  Given a filtration on the space of maps one has the filtration on the group,$G^{(\bullet)}$, and on the tangent space, $T_{G^{(\bullet)}}$. We establishthe criteria of type "$T_{G^{(j)}}f$ vs $G^{(j)} f$" in their strongest form,for arbitrary base field/ring, provided the characteristic is zero or high fora given $f$.  As an auxiliary step, important on its own, we study the mixed modulestructure of the tangent space $T_{A}f$ and establish several properties of theannihilator ideal $Ann[T^1_{A}f]$.

Detecting modules inside the group-orbits R, K, A$\circlearrowright Maps(X,(k^p,o))$. The case of arbitrary characteristic