Let $K$ be an algebraic number field with ring of integers $O_{K}$ and let $G$ be a finite group. We denote by $\mathrm{R}( O_K[G])$ the set of classes in the locally free class group $\mathrm{Cl}( O_K[G])$ realisable by rings of integers in tamely ramified $G$-Galois $K$-algebras. McCulloh showed that, for every $G$, the set $\mathrm{R}( O_K[G])$ is contained in the so-called Stickelberger subgroup $\mathrm{St}( O_K[G])$ of $\mathrm{Cl}( O_K[G])$.

In this paper first we describe the relation between $\mathrm{St}( O_K[G])$ and $\mathrm{Cl}^{\circ }(O_{K}[G])$, where $\mathrm{Cl}^{\circ }(O_{K}[G])$ is the kernel of the morphism $\mathrm{Cl}( O_K[G])\longrightarrow \mathrm{Cl}(O_K)$, induced by the augmentation map $ O_K[G]\longrightarrow O_K$. Then, as an example of computation of $\mathrm{St}( O_K[G])$, we show, just using its definition, that $\mathrm{St}(\mathbb{Z}[G])$ is trivial, when $G$ is a cyclic group of order $p$ or a dihedral group of order $2p$, where $p$ is an odd prime number.

Finally we prove that $\mathrm{St}( O_K[G])$ has good functorial behaviour under change of the base field. This has the interesting consequence that, given an algebraic number field $L$, if $N$ is a tame Galois $L$-algebra with Galois group $G$ and $\mathrm{St}( O_K[G])$ is known to be trivial for some subfield $K$ of $L$, then $O_N$ is stably free as an $ O_K[G]$-module.