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Let $E$ be a generalized cohomology theory. Let's agree that $E$ satisfies property $(*)$ if for any two finite CW complexes with isomorphic homotopy groups their $E$-localizations also have isomorphic homotopy groups. Is there some non-tautological condition implying property $(*)$?

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    $\begingroup$ The only localization I know with this property is rationalization, and of course there is for trivial reasons. $\endgroup$ – Denis Nardin Mar 4 at 6:12
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    $\begingroup$ @AknazarKazhymurat There is a variant of p-localization/completion for non simply connected spaces, but I doubt it has the property in question (I'm not 100% sure what it does to the fundamental group, but it is quite destructive) $\endgroup$ – Denis Nardin Mar 4 at 10:42
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The question is very unlikely to have a satisfactory answer in the level of generality asked. Let $X$ and $Y$ be the two spaces with isomorphic homotopy groups, and let $LX$ and $LY$ denote their localizations with respect to $E$. For certain generalized cohomology theories $E$, we have formulas for $\pi_n(LX)$ and $\pi_n(LY)$, if $X$ and $Y$ are nilpotent, going back to Bousfield's paper The localization of spaces with respect to homology. An excellent summary is provided by Dwyer, and teaches us, for example, that for $E = \mathbb{Z}/p$, one has $\pi_n(LX) \cong \pi_n(X) \otimes \mathbb{Z}_p$, and for any subring $R$ of $\mathbb{Q}$, taking $E = HR$ gives $\pi_n(LX) \cong \pi_n(X) \otimes R$. However, this is only with an assumption about $\pi_1(X)$ (i.e. that $X$ is simply connected, or at least nilpotent). Bousfield's 1997 paper Homotopical Localizations of Spaces gets a similar result for nullification with respect to a Moore space (see Theorem 7.5).

A definitive modern treatment is the book More Concise Algebraic Topology by May and Ponto. On page 395, it says "There are several different ways to generalize to non-nilpotent spaces, none of them well understood calculationally", and later on page 409 says "Little is known about the behavior on homotopy groups of localizations and completions of non-nilpotent spaces."

So, basically, if the OP restricts attention to nilpotent spaces and nicely behaved $E$, then everything should be fine, but it is extremely unlikely to get an answer for general $E$. For counterexamples of what can go wrong for specific $E$, you can look to work of Carles Casacuberta (e.g. Example 4.5 here, or the many examples and counterexamples here).

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