The answer is yes, and in fact the following more general result holds.
Theorem. Let $\mathbf{a}, \, \mathbf{b}, \, \mathbf{c}$ be three recursively enumerable degrees of unsolvability (i.e., Turing degrees) with $\mathbf{a} \leq \mathbf{b}$ and $\mathbf{a} \leq \mathbf{c}$. Then there exists a finitely presented group $L$ such that
- the word problem for $L$ is of degree $\mathbf{a};$
- the power problem for $L$ is of degree $\mathbf{b};$
- the order problem for $L$ is of degree $\mathbf{c}.$
See
D. J. Collins: The word, power and order problems in finitely presented groups, in Studies in Logic and Fundations of Mathematics 71 (Word Problems), North-Holland (1973).