Suppose we have a group $G(T,+)$. It naturally supports "multiplication by an integer", e.g. $5 \cdot t = t + t + t + t + t$ for any $t \in T$.
However, this can't be expanded to $Q$ or $R$: there is no easy way to define what $2.5 \cdot t$ means. But I can define some operation $m: R \times T \to T$ to support that, getting $(G(T, +), m)$.
Is there a name for such pair?
For example, $m$ can be "stretching" for time duration, or plain multiplication with rounding for integers. Possible applications: calculating averages and percentiles.
UPD: desired properties of $m$:
$m(0, t) = \theta $
$m(1, t) = t $
$m(2, t) = t + t$
$m(b, m(a, t)) = m(ab, t)$
A group action of $\mathbb R$ on a set $X$ (ie a map $\varphi:\mathbb R\times X\to X$ such that $\varphi(t,\varphi(s,x)) = \varphi(s+t,x)$ and $\varphi(0,x) = x$) is called a flow.
The situation OP is describing seems to be a set equipped with a group structure and a flow which are compatible in that for any integer $k\in \mathbb Z$, $\varphi(k,x) = x^k$. I don't know that such structures have been explicitly studied and I don't think they have a specific name.