Transformations in Grassmann coordinates on a supermanifold are non-linear, in general. They can be 'linearized' giving rise to a series of $k$-fold vector bundles $Vb_k(M)$, $k=1, 2, 3, \ldots$, associated with a supermanifold $M$ which can be seen as linear approximations of $M$ (up to order $k$). On the other hand we construct the cover functor $F_k$ which takes a supermanifold $M$ to a non-negatively $Z$-graded supermanifold. Both functors, $Vb_k$ and $F_k$, are related by means of the diagonalization functor studied before in [BGR]. If $M$ is a Lie supergroup then the cover of $M$ is a $Z$-graded Lie supergroup the structure of which will be discussed. This work was inspired by a cooperation with E. Vishnyakova.

[BGR] A. Bruce, J. Grabowski, M. Rotkiewicz, Polarisation of graded bundles, SIGMA 12 (2016).