Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic fibration (with possibly no section). Assume that there is a $k$-section of $f$, then a fiber and the $k$-section generate a sublattice $L\subset H^{2}(S,\mathbb{Z})$, which is isomorphic to $U(k)$ (hyperbolic lattice multiplied by $k \in \mathbb{N}$).

Assume also that there is an involution $\sigma$ of $S$ such that induced action $\sigma^{*}$ acts as $-id$ on $L$ (especially preserves $L$). Is it true that $\sigma$ preserves the fibration and acts as $-id$ (of elliptic curve) on each smooth fiber of $f$?

If this is not true, what additional condition is required?

Thanks in advance,