For a vector spaces it always holds that any set of vectors spanning vector space $V$ has a subset of vectors which is a basis for $V$. While for lattices it is not true. For example consider one dimensional lattice spanned by $2,3$ then this lattice is $\mathbb{Z}$ and it is not spanned by any one of vectors. Simply stated my question is the following. Let $L$ be a lattice generated by an orbit of some vector $v\in V$ of dimension $k$. Does it always possible to find subset of the orbit which is a basis for lattice?

More formally: Let $G$ be a finite group. Let $\rho:G\rightarrow GL(V)$ be a representation such that $\rho(g)$ is an integer matrix for every $g$. Let $w\in V$ consider a lattice $$L= span_{\mathbb{Z}} ( {\rho(g)w: g\in G }). $$ Does it always possible to find $g_1,g_2,\ldots g_k$, where $k$ is the dimension of $L$ such that $( \rho(g_i)w)_{i=1}^k$ is a basis over $\mathbb{Z}$ for $L$?