Let $X$ be a pointed topological space equipped with two different actions of $E_\infty$-operad. Each action provides a collection of deloopings $X_i$, where $X_0 = X$ and $\Omega X_i$ is homotopy equivalent to $X_{i-1}$, so there are two $\Omega$-spectra having $X$ as a zeroth space. Can these spectra have different cohomology or different cohomology operations? I would like an explicit example or just a better reason than "why not".

I don't have any applications for such spectra. This question is motivated by thinking about D. Rector's paper Loop Structures on the Homotopy Type of $S^3$, which produces uncountably many different deloopings of three-dimensional sphere. They can be distinguished using cohomology operations.

Let $X$ be a connected pointed topological space equipped with two different actions of $E_\infty$-operad. Each action provides a collection of deloopings $X_i$, where $X_0 = X$ and $\Omega X_i$ is homotopy equivalent to $X_{i-1}$, so there are two $\Omega$-spectra having $X$ as a zeroth space. Can these spectra have different cohomology or different cohomology operations? I would like an explicit example or just a better reason than "why not".

I don't have any applications for such spectra. This question is motivated by thinking about D. Rector's paper Loop Structures on the Homotopy Type of $S^3$, which produces uncountably many different deloopings of three-dimensional sphere. They can be distinguished using cohomology operations.