How can a fraction be a Integer, if the fraction is not equal to a whole number?

It depends on your definitions.

Some authors define "whole numbers" as non-negative integers. By that definition, any negative integer is not a whole number, so any fraction that is equal to a negative integer will not be equal to a whole number.

Whether you consider -2/1 or -4/2 to be a fraction or an integer is between you and your text author. These have the same integer value (-2), but their form is that of a fraction, not an integer. If you consider whole numbers to be non-negative, then neither is a whole number in any event. So, depending on your definitions, -2/1 is a fraction that is an integer, but not a whole number.

A fraction is strictly an ordered pair of integers, (m, n) say, in which the criterion for equality, say (m, n) = (m', n') is mn' = nm'.The set of such ordered pairs contains a subset {(m, 1)} which is isomorphic to the iset of ntegers, and in an abstract sense is indistinguishable from them. This is the justification for writing (m, 1) (usually written m/1) = m.

Some people define "whole numbers" as the set "0, 1, 2, 3, 4, 5, ...".

With that definition -7/1 and -12/6 are fractions which are integers but not whole numbers.