Let $k\in \mathbb{N}$ be a natural number and $p$ be a prime number with $p\nmid k$. (We thank Prof. Bartel for his comment on the latter non divisibility condition.) We denote by $U(p)=\{1,2,\ldots,p-1\}$ the mod-$p$ multiplicative group. Then we have a natural group homomorphism: $\phi:\mathbb{Z}\to U(p)$ which is the unique extension of semi group homomorphism $n\mapsto k^n \pmod p$ defined on semigroup of non negative integers.

Question: Is there a group $G$ containing $U(p)$ such that we have an extension $\psi:\mathbb{Q} \to G$ of $\phi$?

If the answer is yes, is there a universal group $G(k,p)$ with the following property:

There is an extension $\psi_{k,p}:\mathbb{Q} \to G_{k,p}$ of $\phi$. Moreover for any other extension $\alpha :\mathbb{Q} \to H\supset U(p)$ of $\phi$, there is a group homomorphism $\beta: G_{k,p} \to H$ with $\alpha =\beta \circ \psi_{k,p}$?