Yes, there is 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
- B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin-New York, 1982. (pp. 294-299)
- A. Mohammadian, A. Erfanian, M. Farrokhi D. G., and B. Wilkens, Triangle-free commuting conjugacy class graphs, J. Group Theory 19 (2016), no. 6, 1049–1061.