You might find the material on the interpretation of $\mathrm{Ext}$ in terms of extensions at Ext and Extensions to be useful. You probably know that $H^n(G,M) = \mathrm{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z},M)$. You can construct a kind of Baer sum on these central series by taking pullbacks and pushouts, which makes the set of central series into a group. This makes sense, given that the method of adding group extensions which puts it in isomorphism with $H^2$ is known as Baer multiplication (c.f. Weiss, Cohomology of Groups).

You might find the material on the interpretation of $\mathrm{Ext}$ in terms of extensions at Ext and Extensions to be useful. You probably know that $H^n(G,M) = \mathrm{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z},M)$ (which is not exactly the same as the relation between $H^2$ and extensions, but it is similar). You might be able to construct a kind of Baer sum on these central series by taking pullbacks and pushouts, which makes the set of central series into a group. This makes sense, given that the method of adding group extensions which puts it in isomorphism with $H^2$ is known as Baer multiplication (c.f. Weiss, Cohomology of Groups).