What is the distribution of $Y$ given that $Y\sim F(\nu_1, \nu_2)\times 2$? Background
I have a variable $X\sim Beta(2,1)$ and another variable $Y=\frac{X}{1-X}$
wikipedia states: 

If $X \sim \operatorname{Beta}(n/2,m/2)$ then $ \frac{m X}{n\left(1-X\right)} \sim \operatorname{F}(n,m)$

From this, I infer that  $Y\sim F(4,2)*2$.
Knowing the parameterization of $X$, I would like to parameterize $Y$ (and not, e.g., $Y/2$). 
Question
Is there a direct transformation?
What I have tried
I ask because I have tried to simulate the transformation, but optimization often fails, and the transformed variable is better fit by  a $\operatorname{logN}$ than an $\operatorname{F}$ distribution.
set.seed(0)
library(MASS)
X <- rbeta(10000, 2,1)
Y <- X/(1-X)
f.fit <- fitdistr(Y, 'f', start = list(df1 = 4, df2 = 2))
logN.fit <- fitdistr(Y, 'lognormal')
AIC(f.fit) < AIC(logN.fit)

 A: Do you mean a direct transform to the $F$ variate?  That would be $Y = X/(2*(1-X))$, given your parameterization of the beta distribution.
Notes on your trials:
1) Your code has an error - it should divide Y by 2, as in 
Y <- X/(2*(1-X))
2) Try comparing Y/2 to its theoretical distribution instead of using AIC, for example, using the Kolmogorov-Smirnov test:
ks.test(Y, pf, df1=4, df2=2)

On my just-completed run, I got a p-value of 0.7991.
Then test the fit to a lognormal, which, if you use the Kolmogorov-Smirnov test can be done by fitting log(Y) to a Normal with identical results (due to the monotonicity of the transform and the fact that the K-S test compares observed and theoretical cumulative distributions):
ks.test(log(Y), pnorm, mean=mean(log(Y)), sd=sd(log(Y)))

On my just-completed run, I got a p-value of 3.553e-15.
Since AIC isn't calibrated, it can't be used as a goodness-of-fit statistic in the same way that true goodness-of-fit test statistics like K-S can.
Edit in response to Abe's clarifying comment:
Y is distributed as an inverted beta distribution, with parameters (2,1), sometimes known as the beta-prime distribution and sometimes as the inverted beta 2 distribution (in those cases, "inverted beta" refers to the dist'n of 1/X).  In math: $Y \thicksim \beta'(2,1)$. A good general reference is Johnson, Kotz, and Balakrishnan, "Continuous Univariate Distributions" Vol. 2. The R package GB2 has the d,p,q,r functions for the beta prime as a special case of the generalized beta (which is what it really has in it); the documentation will make clear what parameters need to be set to what values.
