  The suspension of the path $P_4$ consists of a $P_4$ and an additional vertex
connected to each of the four vertices, and is denoted by $\hat{P_4}$. The
largest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denoted
by $ex(n,K_3,\hat{P_4})$. Mubayi and Mukherjee in 2020 showed that $
ex(n,K_3,\hat{P_4})= n^2/8+O(n)$. We show that for sufficiently large $n$,
$ex(n,K_3,\hat{P_4})=\lfloor n^2/8\rfloor$.
