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Pre-Image Resistance of repeated, patterned SHA2-256 digests

Suppose $H = operatorname{SHA2-256}$.

Let $H_1 = H(text{“Alpha1”})$.

Let $H_2 = H(text{“Alpha2”})$.

Let $H_N = H(text{“AlphaN”})$.

So all these digests are based off a common pre-image post-fixed with the index.

Question 1: Does knowing the values $H_1, dotsc, H_N$ and the length of common pre-image substring “Alpha” leak any information about “Alpha”?

Question 2: Would double-hashing make any significant difference to pre-image security? E.g.: $H_i = H(H(text{“Alpha”} mathbin| i))$

Pre-Image Resistance of repeated, patterned SHA2-256 digests

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