How to find the number of rabbits after an elapsed time if they increase each time they mate? The problem is as follows:

Jenny is given by her dad as a birthday gift two female rabbits and
two male rabbits. Six months later each female rabbit gives birth
three female rabbits and three male rabbits. If we assume that the
same happens each six months. How many rabbits as maximum will she
have when it has elapsed two years after received them as birthday
present?

The alternatives in my book are as follows
$\begin{array}{ll}
1.&amp;\textrm{1024}\\
2.&amp;\textrm{2048}\\
3.&amp;\textrm{512}\\
4.&amp;\textrm{2192}\\
\end{array}$
I often get confused in these sorts of problems. Can someone help me?. Is this related with the famous Fibonnaci sequence
I think the method which would help me the most is an approach which develops from step by step. The part where I'm stuck is how to account for the female rabbits already born? do they mate as well?. This is kind of confusing. Is there powers involved?.
From looking at the alternatives I can see that there are suspiciously powers of two. Why? what's the reason for this?. Because of this part, it would be very helpful for easing my understanding, explaining the basis of the solution.
I think let's suppose if I have two rabbits, and each one makes two, then in the next generation I will have four, and in the next eight, but I don't have an idea how to use this into the problem. All and all, can someone help me to better understand this?
Every six months, the number of rabbits of either gender gets quadrupled. Hence, the number of rabbits of either gender is $2\cdot4^{n}$ at the end of $n$ half years. Total rabbits$=2\cdot2\cdot4^n=4^{n+1}$. For your question, $n=4$.
For this solution the following is true:

The population of male rabbits is always the same as the population of female rabbits. This is because every female rabbit gives birth to an equal amount of male and female rabbits and we start with an equal amount of males and females.
Males have no direct influence on the growth of the population. This is because males don't give birth
Rabbits never die and females give birth even after they've given it once before.

Since the population of female and male rabbits stays equal and males don't give birth we can just calculate the population of female rabbits and then double it at the end to get the complete population.
The key idea
In the problem it is stated that one female gives birth to 3 other females. This concludes that the the number of new borns is 3 times bigger then the population before 6 months.
Since the old population lives on and gives birth to new rabbits we have to add them to the new borns
Calculating the result
We can transform this into a recursive function $f(t)$ where every $t$ represents $6$ months.
($t=1 =&gt; 6months;$ $t=2 =&gt; 12months...$) :
$p(t+1) = p(t) + 3*p(t)$
The starting is two females so at $t=0$ the population is $2$:
$p(0) = 2$
This gives us the following results:
$p(1) = p(0) + 2*3     = 8$
$p(2) = p(1) + 8*3     = 32$
$p(3) = p(2) + 32*3   = 128$
$p(4) = p(3) +  128*3   = 512$
...
This however is only the population of female rabbits. As I've stated in the beginning we just have to double the population to also include the male rabbits.
This gives us:
$2* p(4) = 2 * 512 = 1024$
So after $4 * 6$ months or $2$ years the population of the rabbits is at $1024$ rabbits.
To explain explain why the population always seems to be a power of 2 we can simply derive the explicit function of the recursive function wich gives us:
$f(t)=2*2*4^t$ or $ f(t)=2*2*(2*2)^t$
Connection to the fibonacii sequence:
As seen in this solution this problem can be solved via a recursive function. The common expression of the FibonaciiSequence is also a recursive function. So they are related in that way.