Extremal $t$-intersecting families for finite sets with $t$-covering number at least $t+2$

Let $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ be a $t$-intersecting family. Define the $t$-covering number ${\tau }_t(\mathcal{F})$ of $\mathcal{F}$ as the minimum size of a subset $S$ of $[n]$ with $\mathrm{\mid }S\cap F\mathrm{\mid }⩾t$ for each $F\in \mathcal{F}$. In this paper, we characterize $\mathcal{F}$ for which $\mathrm{\mid }\mathcal{F}\mathrm{\mid }$ takes the maximum value under the condition that ${\tau }_t(\mathcal{F})⩾t+2$ and $n$ is sufficiently large, thereby generalizing two results by Frankl.