Background: Given a finite group $G$ and a prime $p$ dividing its order, Brauer theory compares the ordinary characters of $G$ with the Brauer characters arising from $p$-modular representations.   On the character level there is a well-defined "reduction mod $p$", which Brauer showed to be surjective in the sense that each irreducible Brauer character lifts to a virtual character ($\mathbb{Z}$-linear combination of ordinary irreducible characters).   Part III of Serre's textbook works this out in the framework of Grothendieck groups.   The lifting is not unique, though J.A. Green's 1955 work <a href="http://www.ams.org/journals/tran/1955-080-02/S0002-9947-1955-0072878-2/">here</a> provides a simple character formula for one lifting.   (This is developed in a more sophisticated way by Lusztig, *Ann. of Math. Studies* No. 81, 1974.) 

An important subtheme, initiated about 70 years ago, concerns blocks (indecomposable 2-sided ideals of the modular group algebra) which have cyclic defect groups: here a defect group is a certain $p$-subgroup of $G$ determined up to conjugacy.   To such a block is associated a *Brauer tree*, a graph with edges labelled by irreducible Brauer characters and vertices labelled by one (or exceptionally several) irreducible ordinary characters.   Chapter VII of Feit's 1982 book has an extensive treatment, while his 1984 paper studies the possible Brauer trees using the classification of finite simple groups.    

For finite groups of Lie type with defining characteristic $p$ (simple or close to simple), the only family giving rise to Brauer trees consists of the groups $\mathrm{SL}(2,p)$ (say for $p>3$).   Leaving aside the Steinberg character (with trivial defect group), there are two blocks with defect group of order $p$.  Consider just the block containing the trivial ordinary (and modular) character, or the corresponding block of $\mathrm{PSL}(2,p)$.   Here the tree is an open polygon with $1_G$ at one end and two exceptional characters at the other end.  The edges correspond to even highest weights $0, p-3. 2, p-5, \dots$.  It's easy to specify Brauer liftings here by following vertices to the left or right with alternating signs.

> For an arbitrary $G$ having a block with a nontrivial Brauer tree, is there a similar algorithm for Brauer lifting (in particular, an algorithm for Green's special lifting)?

I don't know all the literature well enough to sort this out, but the idea would be somewhat in the spirit of Green's "walk around the Brauer tree" (*J. Austral. Math. Soc.* 17, 1974) for producing a projective resolution (necessarily infinite, but periodic).  For this purpose the Brauer tree simultaneously encodes the projective covers needed, via Brauer reciprocity.