If $f:2^{\omega}\longrightarrow 2^{\omega}$ is a homeomorphism and $J\subseteq 2^{\omega}$ is a maximal ideal, then $f[J]$ is a maximal ideal?

Question

Working on a problem about restrictions of homeomorphisms to maximal ideals of the set of Cantor $2^{\omega}$ I came up with the following question:

If $f:2^{\omega}\longrightarrow 2^{\omega}$ is a homeomorphism and $J\subseteq 2^{\omega}$ is a maximal ideal, then $f[J]$ is a maximal ideal?