I'm looking at Brunner's version of Adam's computation showing $H^*(ku;\Bbb Z_2)=\mathcal A/\mathcal A(Q_0,Q_1)$. Here $ku$ is the connective cover of the usual complex $K$-theory spectrum $K=KU$. The first several spaces are $$ku=(BU\times\Bbb Z,U,BU,SU,BSU,BSU\langle 5\rangle,\ldots).$$ Bott periodicity shows $\pi_*ku=\Bbb Z[\eta]$, where $\eta$ has degree $2$ and corresponds to the Hopf map. From this we can construct a Whitehead tower for $ku$:
\begin{array}{ccccc} & & \vdots & &\\ & & \downarrow & & \\ \Sigma H\Bbb Z &\longrightarrow & \Sigma^4 ku &\longrightarrow & \Sigma^4 H\Bbb Z\\ & & \phantom{\eta}\downarrow\eta & & \\ \Sigma^{-1}H\Bbb Z&\longrightarrow & \Sigma^2 ku &\longrightarrow & \Sigma^2H\Bbb Z \\ & & \phantom{\eta}\downarrow\eta & & \\ & & ku& \longrightarrow & H\Bbb Z \end{array}
Now my understanding is that the $0$-th $k$-invariant is classified by $ku\to H\Bbb Z$ and lives in $H^0ku$. This would just be $k_0=1\in H^0ku$. The next $k$-invariant should be classified by $\Sigma^2ku\to \Sigma^2H\Bbb Z$, giving a class in $H^2\Sigma^2ku$.
But Bruner refers to the first $k$-invariant as (induced by) the composition $$\Sigma^{-1}H\Bbb Z\to \Sigma^2ku\to\Sigma^2H\Bbb Z,$$ and thus an element of $H^2(\Sigma^{-1}H\Bbb Z)=H^3H\Bbb Z$. I cannot see why it makes sense to call this "the" first $k$-invariant. Adams uses the same language, simply asserting that the first $k$-invariant of $ku$ lives in $H^3H\Bbb Z$. I feel like I'm missing something obvious here and would appreciate some explanation or reference.