For a filter $\mathcal F$, $S_{\mathcal F}=\{\infty\}\cup\{(n,m): n,m\in\mathbb N\}$ be the $\mathcal F$-hedgehog ($\mathcal F$-fan) of spininess $\omega$ where each $(n,m)$ is isolated in $S_{\mathcal F}$ and a basic open neighborhood of $\infty$ is of the form $N(\varphi)=\{\infty\}\cup\{(n,m):n\in\mathbb N,m\in\varphi(n)\}$ for function $\varphi\colon\mathbb N\to\mathcal F$. We study some connections among the $\mathcal F^*$-Menger property and an embedding of $\mathcal F$-hedgehog $S_{\mathcal F}$ into function spaces for any $P$-filter $\mathcal F$.