In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.

There exist $\Sigma$-free operads $\mathcal{C}_n$, $1\le n\le \infty$, such that every n-fold loop space is a $\mathcal{C}_n$-space and every connected $\mathcal{C}_n$-space has the weak homotopy type of an $n$-fold loop space.

For $\mathcal{C}_n$ the explicit example constructed in Peter May's book is the operad of little $n$-cubes.

However, there is a stronger version of the recognition theorem for **grouplike** $\mathcal{C}_n$-spaces spread all over the literature (e.g. at the nLab).

Although I'm absolutely not an expert in this field, I believe it should not be hard to prove the stronger form with the help of the techniques developed in May's book as soon as one dwelled on. (Is this the case? Is there a reason May did not state and prove the stronger version?)

Is there a reference for the stronger version that one can use if one does not want to expect the readers to be capable extending May's proof? I don't want to expect any knowledge of higher category theory, so Lurie's text on $E_k$-algebra does not fulfill my needs.