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Let $\lambda$ and $\mu$ be solid sequence spaces. For a sequence of modulus functions $\Phi =\left({\varphi }_k\right)$ let $\lambda \left(\Phi \right)=\{x=\left(x_k\right)({\varphi }_k\left(\right|x_k\left|\right))\in \lambda \}$. Given another sequence of modulus functions $\Psi =\left({\psi }_k\right)$, we characterize the continuity of the superposition operators $P_f$ from $\lambda \left(\Phi \right)$ into $\mu \left(\Psi \right)$ for some Banach sequence spaces $\lambda$ and $\mu$ under the assumptions that the moduli ${\varphi }_k$ $(k\in \mathbb{N})$ are unbounded and the topologies on the sequence spaces $\lambda \left(\Phi \right)$ and $\mu \left(\Psi \right)$ are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type....