Properties of the functor which changes the topological subgroup $\mathbb R^\delta$ to $\mathbb R$

Consider the pushout functor
$$ \mathbb R *_{\mathbb R^\delta} {-} $$
from the under-category $\mathbb R^\delta$-TGrp of topological groups under $\mathbb R$ with discrete topology to the under-category $\mathbb R$-TGrp of topological groups under $\mathbb R$. We will investigate basic properties of this functor, such as homotopy properties and the presentation of $\mathbb R *_{\mathbb R^\delta} X$ as the colimit of a sequence $F^1(X)\to F^2(X) \to \dots$.
We also state the problem of the calculation of the cohomology groups
$$ H^* (\mathbb R *_{\mathbb R^\delta} X;\mathbb{F} _p) $$
and provide this calculation for central and normal morphisms $\mathbb R^\delta\to X$.