This should be possible for any group, assuming I have facts about extending measures correct. The idea is to first construct such a measure on a smaller Boolean algebra of subsets of $G$, and then use general measure theory facts to extend this measure to $\mathcal{P}(G)$.

So first, let $\mathcal{B}\subseteq\mathcal{P}(G)$ be the Boolean algebra generated by cosets of finite index subgroups of $G$. Define a measure $\mu$ on $\mathcal{B}$ as follows. Suppose $A\in\mathcal{B}$. Then there is some finite index subgroup $H$ of $G$ such that $A$ is a union of cosets of $H$. If $n$ is the index of $H$, and $m$ is the number of cosets in the union, then define $\mu(A)=m/n$. It can be directly checked that $\mu$ is well-defined. Moreover, if $C$ is a coset of subgroup of index $n<\infty$, then $\mu(C)=1/n$ by construction. Note that $\mu$ is also translation-invariant.

Now one can (non-uniquely) extend $\mu$ to some finitely additive measure on $\mathcal{P}(G)$ (which won't be translation-invariant necessarily). I believe this follows from Section 457 of Fremlin's "Measure Theory". Specifically, any finitely additive probability measure on a Boolean algebra (of subsets of some fixed set $X$) can be extended to such a measure on any larger Boolean algebra.

**Remark 1:** The initial measure $\mu$ is in fact the unique $G$-invariant finitely additive probability measure on $\mathcal{B}$, and can be constructed from the Haar measure on the profinite completion of $G$. In particular, if $\mathcal{N}$ denotes the collection of finite index normal subgroups of $G$, then the profinite completion is $\hat{G}=\varprojlim_{\mathcal{N}}G/N$. We can write elements of $\hat{G}$ as $(C_N)_{N\in\mathcal{N}}$, where $C_N$ is a coset of $N$. Given a set $A$ in $\mathcal{B}$, define $X_A$ to be the set of $(C_N)_{N\in\mathcal{N}}\in \hat{G}$ such that $C_N\cap A\neq\emptyset$ for all $N\in\mathcal{N}$. Then $X_A$ is closed, and it can be shown that $\mu(A)$ is the Haar measure of $X_A$.

**Remark 2:** Perhaps it's also worth mentioning that if $G$ is amenable (e.g., abelian) then there is a translation-invariant finitely additive probability measure on $\mathcal{P}(G)$, which must satisfy the desired conditions outright by finite additivity and invariance.

everysubgroup of index $n$? For that matter, is $n$ fixed, or is this meant to hold for every finite-index subgroup? $\endgroup$ – LSpice Jun 17 '19 at 22:19