Hecke operators in Morava $E$-theories of different heights

There is a natural action of a kind of Hecke algebra $\mathcal{H}_n$ on the $n$th Morava $E$-theory of spaces. We construct Hecke operators in an amalgamated cohomology theory of the $n$th and the $(n+1)$st Morava $E$-theories. These operations are natural extensions of the Hecke operators in the $(n+1)$st Morava $E$-theory, and they induce an action of the Hecke algebra $\mathcal{H}_{n+1}$ on the $n$th Morava $E$-theory of spaces. We study a relationship between the actions of the Hecke algebras $\mathcal{H}_n$ and $\mathcal{H}_{n+1}$ on the $n$th Morava $E$-theory, and show that the $\mathcal{H}_{n+1}$-module structure is obtained from the $\mathcal{H}_n$-module structure by the restriction along an algebra homomorphism from $\mathcal{H}_{n+1}$ to $\mathcal{H}_n$.