TITLE: Analytic function avoiding elements of the modular group
QUESTION [10 upvotes]: A friend recently told me the following two facts, for which he cannot recall a proof or a reference
(but he remembers seeing them in the literature):

Let $f$ be a holomorphic function mapping the upper half-plane $H$ into itself. Let $G$ be the
group of fractional linear transformations $(az+b)/(cz+d)$ where $ad-bc=1$, $a,d$ are odd integers
and $b,c$ are even integers. Suppose that for every $g\in G$ and for every $z\in H$,
$f(z)$ is not equal to $g(z)$. Then $f$ is fractional-linear. (Or maybe such $f$ just does not exist).
Let $f$ be the same as before. Suppose that for all integers $m,n$ and all $z\in H$
we have $f(z)\neq mz+n$. Then $f$ is fractional-linear.

I will appreciate any relevant reference or any other information. 
EDIT: There is no $f\in Aut(H)$ that satisfies the condition of Problem 1. This implies that $f$ constructed by
 Aakumadula is NOT fractional-linear.
To prove this, we write $(az+b)/(cz+d)=(xz+y)/(uz+t)$, where $a,b,c,d$ are given real numbers,
and we want to find integers $x,y,u,t$, where $x,t$ are odd, and $y,u$ are even, so that this has non-real roots
$z$. This is to show that certain quadratic form in $a,b,c,d$ is indefinite. And this is performed
by an elementary calculation.

REPLY [6 votes]: Concerning question 2 see  Earle, Clifford J. On holomorphic families of pointed Riemann surfaces. Bull. Amer. Math. Soc. 79 (1973), 163–166 and Theorem 3 there.