We consider the class of $\mathbb{R}^n$ extensions of Anosov diffeomorphisms on infranilmanifolds, and find necessary and sufficient conditions for topological transitivity. In particular, if the fiber is $\mathbb{R}$, the existence of a semi-orbit with the projection on $\mathbb{R}$ unbounded from above and from below is equivalent to topological transitivity. We also show that in the above class topological transitivity and stable topological transitivity are equivalent.