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

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

Indeed, the same results holds for all primes $p$ and automorphisms $\theta:x\mapsto x^p$ of $GF(p^n)$.

See

1. B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin-New York, 1982. (pp. 294-299; also pp 299-316 for supplementary results)
2. 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.

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

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

Indeed, the same results hold for all primes $p$ and automorphisms $\theta:x\mapsto x^p$ of $GF(p^n)$.

See

1. B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin-New York, 1982. (pp. 294-299; also pp 299-316 for supplementary results)
2. 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.

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 Suzuki $2$-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, Springer-Verlag, Berlin-New York, 1982. (pp. 294-299; also pp 299-316 for supplementary results)
2. 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.

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

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

Indeed, the same results holds for all primes $p$ and automorphisms $\theta:x\mapsto x^p$ of $GF(p^n)$.

See

1. B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin-New York, 1982. (pp. 294-299; also pp 299-316 for supplementary results)
2. 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.

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

1. B. Huppert and N. Blackburn, Finite Groups II, 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, J. Group Theory 19 (2016), no. 6, 1049–1061.

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 Suzuki $2$-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, Springer-Verlag, Berlin-New York, 1982. (pp. 294-299; also pp 299-316 for supplementary results)
2. 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.