It seems that we have here a problem of interpretation (and thus also a problem with translation).
See Aristotle, Nicomachean Ethics (R.Crisp editor, 2000), Introduction, page xxi:
The subject of Plato's Republic is what in Greek is called dikaiosynē.
This word is usually translated "justice", but in a recent translation the
word "morality" is used. This choice reflects an ambiguity in the Greek,
itself implicit in Aristotle's distinction between general and particular
justice in the first two chapters of book V. [...] Aristotle ties this conception of complete virtue to the law, his thought being that the law (ideally speaking) aims ultimately at the instantiation of all the virtues in the citizens it governs. So what is generally just is what is lawful.
See Book V :
[1129a] We see that everyone means by justice the same kind of state, namely, that which disposes people to do just actions, act justly, and wish for what is just.
[1129b] Let us acquire some grasp, then, of how many ways there are in which
a person is said to be unjust. Both the lawless person and the greedy and
unfair person seem to be unjust. Obviously, then, both the lawful person
and the fair person will be just; and thus the just is the lawful and the
fair, and the unjust is the lawless and the unfair. [...] Since, as we saw, the lawless person is unjust and the lawful just, it is clear that whatever is lawful is in some way just; for the things laid down by legislative science are lawful, and each of these we describe as just [emphasis added; this is - of course - a debatable assumption: Aristotle states that laws are, per se, "moral"].
[1130b] Clearly, then, there are several kinds of justice, and there is one that is distinct from virtue as a whole; we must ascertain what it is and what
sort of thing it is. What is unjust has been divided into what is unlawful and what is unfair, and what is just into what is lawful and what is fair. Injustice in the sense above corresponds to what is unlawful. But what is unfair is not the same as what is unlawful, but differs as part from whole (since everything that is unlawful is unfair, while not everything that is unfair is unlawful [emphasis added; here the translation differs from yours and other ones; so, the issue is : which is the "correct" one ? ]); and so what is unjust, and injustice in the sense of unfairness, are not the same as what is unjust and injustice in the other sense, but differ as parts from wholes.
The point lies for me in the "ambiguity" of justice; we have that:
"the just is the lawful and the fair, and the unjust is the lawless and the unfair [1129b]."
According to this statement, fairness [equality] is "included" into justice, and so injustice is "included" into unfairness.
See Aristotle's Political Theory :
Aristotle's constitutional theory is based on his theory of justice, which is expounded in Nicomachean Ethics; Book V. Aristotle distinguishes two different but related senses of “justice” — universal and particular — both of which play an important role in his constitutional theory. Firstly, in the universal sense “justice” means “lawfulness” and is concerned with the common advantage and happiness of the political community. [...] Secondly, in the particular sense “justice” means “equality” or “fairness”, and this includes distributive justice, according to which different individuals have just claims to shares of some common asset such as property.
In 1130b, Aristotle says: "what is unjust, and injustice in the sense of unfairness [the second sense of "justice"], are not the same as what is unjust and injustice in the other sense [the first sense : unlawful], but differ as parts from wholes."
Here we have a mismatch: fair is included into justice (the "general" concept) but can be distinct from lawful. Thus, if we consider a fair action that is "outside" law, its contrary will be an unfair one that can still be consistent with law.
This means that, in general, unfairness is not included into unlawful.
Thus, if we equate justice with "acting according to law" [the first sense of "justice"], we may have some cases of "unfair acting" that are not injust, i.e. they are not "acts contrary to law".
Consider now the following example:
[1131a] One type of particular justice, and of what is just in that same sense, is that found in distributions of honour or money or the other things that have to be shared among members of the political community (since here one person can have a share equal or unequal to another's).
This makes sense if we consider justice in the sense: "according to law".
In a similar vein, we may imagine someone leaving his money after his death to his two sons, but dividing it into unequal parts. If there is no law forbidding it, this act is "lawful" but from a moral point of view we can call it "unfair".