Yes, there an infinite series of these groups. Let 
$$A(n):=\left\{\begin{bmatrix}1&0&0\\a&1&0\\b&a^2&1\end{bmatrix}:a,b\in GF(2^n)\right\}.$$
be a group of order $2^{2n}$ for $n\geq3$.

Then $G/\Phi(G)\cong\Phi(G)\cong C_2^n$ and $\{1,\Phi(G),G\}$ is the set of all  characteristic subgroups of $G$.

See

 1. B. Huppert and N. Blackburn, [*Finite Groups II*](https://link.springer.com/book/10.1007/978-3-642-67994-0), Springer-Verlag, Berlin-New York, 1982. (pp. 294-299)
 2. A. Mohammadian, A. Erfanian, M. Farrokhi D. G., and B. Wilkens, [Triangle-free commuting conjugacy class graphs](https://doi.org/10.1515/jgth-2016-0002), *J. Group Theory* **19** (2016), no. 6, 1049–1061.