Bases of associated Galois modules in general wildly ramified extensions and in elementary abelian extensions of degree $p^2$

For a wildly ramified extension $K\mathrm{/}k$ of complete discrete valuation fields we study collections of elements of $k[G]$ (where $G=Gal(K\mathrm{/}k)$) that fit well for constructing bases of various associated Galois modules and orders. In the case $G=(Z\mathrm{/}pZ{)}^2$ (where $p$ is the characteristic of residue fields) we are able to compute the action of the elements $({\sigma }_1-1{)}^i({\sigma }_2-1{)}^j,0\le i,j\le p-1,$ on the valuation filtration; here ${\sigma }_1,{\sigma }_2$ are generators of $G$. If the ramification jumps of $K\mathrm{/}k$ are distinct modulo $p^2$ then these elements do yield “good enough” bases in question.