  In this article we construct a family of genus two Lefschetz fibrations
$f_{n}: X_{\theta_n} \rightarrow \mathbb{S}^{2}$ with $e(X_{\theta_n})=11$,
$b^{+}_{2}(X_{\theta_n})=1$, and $c_1^{2}(X_{\theta_n})=1$ by applying a single
lantern substitution to the twisted fiber sums of Matsumoto's genus two
Lefschetz fibration over $\mathbb{S}^2$. Moreover, we compute the fundamental
group of $X_{\theta_n}$ and show that it is isomorphic to the trivial group if
$n = -3$ or $-1$, $\mathbb{Z}$ if $n =-2$, and $\mathbb{Z}_{|n+2|}$ for all
integers $n\neq -3, -2, -1$. Also, we prove that our fibrations admit $-2$
section, show that their total space are symplectically minimal, and have the
symplectic Kodaira dimension $\kappa = 2$. In addition, using the techniques
developed in \cite{A, AP1, ABP, AP2, AZ, AO}, we also construct the genus two
Lefschetz fibrations over $\mathbb{S}^2$ with $c_1^{2} = 1, 2$ and $\chi = 1$
via the fiber sums of Matsumoto's and Xiao's genus two Lefschetz fibrations,
and present some applications in constructing exotic smooth structures on small
$4$-manifolds with $b^{+}_{2} = 1$ and $b^{+}_{2} = 3$.
