Suspension homotopy classes have natural group structures

Represent a suspension map by $a(x,s)$ constant at the basepoint for s=0,1 and x=x0. Define $(a\ast b)(x,s)=a(x,2s)$ on the first half and $b(x,2s-1)$ on the second. The values match at the seam, and the same pasting on homotopies proves representative independence. F2 descends each pasted map and homotopy to the suspension.

Apply the explicit homotopies of F3 with x left as a parameter: $a(x,(1-t)s+t\varphi (s))$ for unit and associativity reparametrizations, and $a(x,2s(1-t))$, $a(x,2(1-s)(1-t))$ on the two halves for reversal cancellation. Their formulas are jointly continuous, preserve both endpoint values and the entire x0 track, and descend by F2. Thus F3 proves associativity, a two-sided unit and a two-sided inverse here.

On double suspensions, both coordinates admit the operation. The four-quarter identity holds pointwise with x unchanged. The two-unit calculation in F3 therefore identifies the operations and proves commutativity. Finally, substituting f(x) for x commutes pointwise with the half-interval formulas, so precomposition by $\mathrm{\Sigma }f$ preserves multiplication and the constant class.