Full groups of one-sided topological Markov shifts

Let $(X_A,\sigma_A)$ be the right one-sided topological Markov shift for an irreducible matrix with entries in $\{0,1\}$, and $\Gamma_A$ the continuous full group of $(X_A,\sigma_A)$. For two irreducible matrices $A$ and $B$ with entries in $\{0,1\}$, it will be proved that the continuous full groups $\Gamma_A$ and $\Gamma_B$ are isomorphic as abstract groups if and only if their one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ are continuously orbit equivalent.