Spectral extremal graphs for $F_6$-free graphs with even size

Let $F_l$ be the fan graph obtained by joining a vertex with a path on $l-1$ vertices. Yu, Li and Peng [Discrete Math. 346 (2023)] conjectured that if the number of edges of $G$ is $m$ and the spectral radius $\lambda (G)>\frac{k-1+\sqrt{4m-k^2+1}}{2}$, then $G$ contains a $F_{2k+1}$ and $F_{2k+2}$, unless $G=K_k\vee (\frac{m}{k}-\frac{k-1}{2})K_1$. The case $k\ge 3$ of the above conjecture has been confirmed by Li, Zhao and Zou [J. Graph theory 110 (2025)]. Zhang and Wang [Discrete Math. 347 (2024)], Yu, Li and Peng [Discrete Math. 348 (2025)], Gao and Li [Discrete Math. 349 (2026)] confirmed the case $k=2$. However, the extremal graphs for the case $k=2$ only exist when $m$ is odd. The case with $m$ even has not been determined. In this paper, we characterize the extremal graph for $F_6$ and even $m\ge 3000$.