When studying the stable homotopy of $BG^{\wedge}_p$, with $G$ a finite group, authors know that this abuse of notation is not dangerous because both $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^{\wedge}_p$ are stable homotopy equivalent. However, that abuse is also common when $G$ is a compact Lie group, I do not know any reference that justifies this abuse unlike the finite group case (in this case $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^{\wedge}_p$ are not always stable homotopy equivalent).
What is more, some result such as the following from a Chun-Nip Lee's article:
Let $Y_1, Y_2$ be connected CW complexes such that both $H^{\ast}(Y_1, \mathbb{F}_p )$ and $H^{\ast}(Y_2, \mathbb{F}_p )$ are finitely generated as graded rings. Then the image of the ring homomorphism $\{(Y_1)^{\wedge}_p,(Y_2)^{\wedge}_p\}\rightarrow\mathrm{Hom}_{\mathbb{F}_p} (\widetilde{H}^{\ast}(Y_1, \mathbb{F}_p ), \widetilde{H}^{\ast}(Y_1, \mathbb{F}_p ))$ is finite.
What did the author mean by $(Y_i)^{\wedge}_p$?. In general, is there a criterion to know whether $\Sigma^{\infty}Y^{\wedge}_p\simeq (\Sigma^{\infty} Y)^{\wedge}_p$ or not?