In Rudyak's book it is proved (3.22 page 160) that every additive cohomology theory (in the sense of Eilenberg-Steenrod) defined on the category of finite dimensional pointed CW complexes can be represented by a spectrum, and this representing spectrum is unique up to equivalence.
This implies the existence and uniqueness of extension of additive cohomology theories from this category to the category of pointed CW complexes.
I am interested in a slightly different setting, and wonder whether the existence and uniqueness of extension still hold:
Question: Let F be a cohomology theory defined on the category of finite dimensional, countable CW complexes, where a cohomology theory for me means an absolute theory satisfying a Mayer-Vietoris property. Is there a unique (up to iso) extension to an additive cohomology theory for CW complexes, and more generally, if the cohomology theory is an equivariant cohomology theory (Z graded, G compact Lie group) defined on the category of finite dimensional countable G-CW complexes, is there a unique (up to iso) extension to an additive cohomology theory for G-CW complexes.