= Solution
\b[False.] On the additive <topological group> $\mathbb R$, put $d(x,y)=|x-y|$ and $d'(x,y)=\sqrt{|x-y|}$. Both are invariant under left and right translations and induce the usual topology. The triangle inequality for $d'$ follows from $\sqrt{a+b}\leq\sqrt a+\sqrt b$ for $a,b\geq0$. But
$$
\frac{d(x,y)}{d'(x,y)}=\sqrt{|x-y|}
$$
can be arbitrarily close to zero and arbitrarily large. Consequently no $K>0$ can satisfy both proposed inequalities. Compatibility of the two invariant metrics is topological equivalence, which need not be <bilipschitz equivalence>.
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