Different locally compact metrizable second countable topologies on the same group

Let $$G$$ be a non-Abelian infinite group. Can $$G$$ admit more than one (inequivalent) non-compact locally compact metrizable second countable topologies that make it a topological group?

Thank you.

Turning my comment into an aswer: yes, it is possible for such a $$G$$ to have more than one topology with your requirements.
Consider $$(\Bbb R,+)$$ with its usual topology. As a group it is isomorphic to $$(\Bbb C,+)$$, so there is also a topology in which $$(\Bbb R,+)$$ is a topological group homeomorphic to $$\Bbb R^2$$, both topologies obviously satisfy your requirements.
The only issue is that $$\Bbb R$$ is Abelian, but that can be fixed by taking a direct product with any locally compact, second countable, metrizable, nonabelian topological group, since all of this properties are preserved under finite products.